There's a pile right there. Okay. All right. So we'll go ahead and move on. I'll kind of let you know what the plan is. So we're going to go on to section 1.2, the binomial theorem. We'll talk about that today. I'm going to give you another assignment. So again, because of Tuesday, they kind of shifted everything back by a day. We'll probably get back at some point soon, hopefully, where I'm going to have you start training assignments in on Tuesday. But the next assignment, then, we'll just be doing a week from today. And so I will be giving you that today. And then we'll kind of do what we did before. And on Tuesday, we'll talk about a couple problems and such and help you get kind of comfortable with this. If we have time today, I'll do a problem or two, but I don't know that we will. So like I said, let's go ahead and get started here. Let's see. Okay. It's just as bad as before. Okay. So as I said, at least we have lines here. This section is called the binomial theorem. You've certainly seen some special cases of this theorem before. If you haven't seen it in full generality, you've seen this before. So, like usual, I'm just going to try to sort of build everything up here from the basics. So we're going to start with a definition. And the book actually defines this in the first section, but I just waited until this section because this is when we're really actually going to start using these things. You've probably all seen this before. Something called a factorial. Okay. So what we're going to do is we're going to make a recursive definition. Okay. And remember, I talked about this word recursion. This is sort of the inductive version of a definition. And we're going to do this on the natural numbers. And it's just going to go like this. This may look a little different than what you're used to, but now that we've seen induction, this really is sort of, in some sense, the best way to make this definition. So one, again, I assume that you've all seen this before. One factorial equals one. And of course I'm going to write down the definition here in a second. How many of you have seen this? The exclamation point thing, okay. Yeah. And then assuming we've defined n factorial, we define n plus one factorial by, okay, n plus one times, and I'll put this in parentheses so there's just no confusion about this, times n factorial. Okay, so n plus one factorial is defined to be n plus one times n factorial. And then this, as I said, this symbol, n with the exclamation point is not a surprising n. This is red n factorial. So there's the definition. And we'll just go through a quick example here. So of course, and I'll explain this as we go through here, but for a natural number n in factorial, it's just the product of all of the natural numbers less than or equal to n. You just multiply them all together, right? Okay, so again, just to make sure that everybody's with me on this. Sorry, this is a little messy. So of course by definition, by one above one factorial is one. And but I want you to actually look at the definition here just because I want you to get used to this because there will be instances where more complicated expressions are going to be defined. Two factorial, of course you might say, oh well it's what you just said, it's two times one. Which it is, but let's just go through the definition and see how we can get that. Okay, so two factorial by definition is of course, just because one plus one is two, this is one plus one factorial, which is equal to, okay, I want you to look at the second definition. We're going to use this in proof. I'm also defining it this way because you really need to be able to understand that n plus one factorial really is n plus one times n factorial. So one plus one factorial then is just plug in, in the second definition, plug in one for n, right? So it's one plus one times one factorial when you plug in one for n, right? So this of course is just two times, I'll just do one more. Three factorial is two plus one factorial, which again, if you look at the second definition, right? This is just two plus one times two factorial, and we just figured out what two factorial was. So this is three times two times one. Of course we don't need the one, but that's what we get. And similarly we can just keep doing this, right? So four factorial is four times three times two times one, and you get the familiar definition that you're used to before you saw me write the inductive one, right? Okay, so we're also going to, I'm singling this out especially, but I could have just built this into the definition too. We're also going to, and you'll see, I think you'll actually appreciate why we're doing this here after this section has been lectured on. We're going to also define zero factorial to be equal to one. Okay, and this is really just done for convenience so that formulas work out without a bunch of caveats before the theorems and such. So it just makes everything uniform in this way. And again, you'll appreciate this. I think once you see the binomial theorem, why we would want to define it this way. Okay, so now we're going to look at sort of the main object of study here in this section. It's called binomial coefficients. And so we're going to suppose that zero is less than or equal to k, which is less than or equal to n. I may suppress this from time to time, but I'll be clear here. k and n are non-negative integers. They're not just arbitrary real numbers. We're still living in the realm of the integers now. So the inequality, of course, implies that they're both non-negative. So all I really need to say here is that k and n are integers, something called the binomial coefficient. And you may wonder why this terminology is coming from. And you'll see here in a minute, once we do the binomial theorem, where this terminology is coming from. So here's the notation. Big left parentheses and above a k, and then a big right parentheses. Okay, so there's a way to read this. Yes, thank you. Good. You've heard of this before. I don't think the author actually says this in the book, which is bizarre. But n choose k. This is what everyone else does, except for the author lives in his own world. But this is read n choose k. There's actually a reason for this, which I won't go into. But let me, in detail anyways, but let me get this down first. I can kind of give you an idea of where this comes from, where this terminology comes from. Okay, so n choose k is equal to n factorial in the numerator divided by k factorial times n minus k factorial. So everybody can see this. Okay, yeah. It's readable, I guess. So let me just say a couple of things about this. First of all, I want you to notice that k itself could be zero. There's nothing precluding k from being zero. But you all learned this a long time ago. You can't divide by zero. But the definition of zero factorial tells us that this denominator is never zero. Even if k is zero, this is not zero. And since k is bigger than or equal to n, n minus k is bigger than or equal to zero. So this factorial is defined for n minus k. And it's because k is less than or equal to n. If k was five and n was two, right? Sorry. So if, yeah, so that works. If k was five and n was two, then you'd have minus three factorial, which we haven't defined. So everything still is perfectly well defined given what we did above, right? So where does this n choose k business come from? Well, does anybody know this? Do you know where this comes from? Yeah. Joe. Right. So the best way to say it really is if you have a set that has n elements in it, then the number of k element subsets is n choose k. That's what this is coming from. So we're not going to go into that aspect of it. We're going to look at it from a different perspective, which is from the point of view of algebra, of multiplying out, taking powers of a sum of two numbers or two variables or two polynomials. It doesn't really matter. But that's sort of the focus. And that's where this binomial coefficient comes from. It's going to come from the theorem. The binomial theorem is dealing with how to find the coefficients in front of a and b when you're multiplying out, say, a plus b to the 50th. What are all the coefficients on the powers of a and b? So that's what we're going to get to here pretty soon. But before I do that, I'll go ahead and do another example just in case you haven't seen this stuff before, just to make sure everybody's with me. Yes, actually it is. You might notice that my writing is a little neater. I had two people come to my office today and both people said, your handwriting is just impeccable on the board. It's amazing. You wouldn't know that from this. If I was writing on the whiteboard, he would say, wow, he has the best handwriting of anyone I've ever seen. But it's slowly improving. Maybe it'll get there at some point. Okay, so let's just look again. This is not meant to be difficult. So this is just a computational example just to make sure that everybody is understanding the ideas here. 8 choose 3. Okay, well, I'm not going to go back to the other slide now, but the definition is that this is 8 factorial, right? divided by 3 factorial times 5 factorial, right? n choose k was n factorial over k factorial times n minus k factorial. The n is 8, the k is 3, so n minus k factorial is 5 factorial in this case, right? Okay, so let's just write this out. I'm going to write out more than what we have to, but okay, so this is 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1, right? Oops, sorry about that. Okay, 3 times 2 times 1 times 5 times 4 times 3 times 2 times 1. Easy enough, right? Just using the definition of factorial. Okay, well, what can we do? Certainly we can do that, right? So just canceling, I'm just canceling out 5 times 4 times 3 times 2 times 1 from the top and the bottom, which of course we can do more than that, right? So we can cancel the 6 and the 3 times 2, and so this just becomes 56. So hopefully that's easy enough. Everybody okay with this, right? Okay, so now we're going to actually start doing some more theoretical calculations here, and I'm just going to call this lemma 1. This is called, this is pretty uniform I think, this is called Pascal's Rule. Okay, and here's what it says. This is actually a lot more useful than you might think right now, and in fact you're going to use this in the exercises quite a bit. So we're going to assume that, I'm going to use a little shorthand here. I'm going to leave out a few things. Okay, so I should be very clear again on this. These factorials are only defined for non-negative integers. So everything, when I say 1 is less than or equal to k, less than or equal to n, I mean that k and n are integers. I'm not going to write that out every time, but it's understood that k and n are integers here, okay? Okay, and so here is the result. So 1 is less than or equal to k, which is less than or equal to n. Then the result is n choose k plus n choose k minus 1 equals n plus 1 choose k. Okay, and actually as it turns out, there's a nice little trick that makes this proof very short, very understandable. And this result is very useful too, and you'll see how this is used. And by the end of the homework, you'll hopefully have a really good appreciation for how nice this result actually is. Okay, so just basic algebra really. So the first thing to note is, okay, let me offset this with an asterisk. Let's take 1 over k and we're going to add 1 over n minus k plus 1. So 1 over k plus 1 over n minus k plus 1. Okay, so you know how to do this, I assume.  Okay, so what did I do here? Common denominator. That's it. That's all I'm doing. Multiply the top and bottom of the first fraction by n minus k plus 1, multiply the top and bottom of the second by k, and that's where this numerator comes from. Okay, so basic algebra is all this is. Okay, and what does this equal to? Well, you notice that we can cancel the k out on the top, right? So we just have n plus 1 divided by k times n minus k plus 1. Okay, so now what we're going to do is we're going to multiply both sides of this equation, n factorial divided by k minus 1 factorial times n minus k factorial. You may wonder where this is coming from, but you'll see actually by doing this that we're going to get the desired equation to fall out pretty quickly. I'm going to go through this slowly to make sure I don't lose anybody. If you just see where this is coming from, the rest of it is very, very simple. And actually this is really pretty easy too. Okay, almost done. This looks a little bit messy, but I want to just remind you that this is not anything that's complicated really. Okay, so the equation, the start equation, all I'm doing is multiplying both sides of that. I'm sort of ignoring the middle, okay? I'm multiplying both sides of it by n factorial over k minus 1 factorial times n minus k factorial. So if you just look at the first term, right, 1 over k, what do we get when we multiply that first term by this? Well, we get n factorial in the numerator, right? You see that? Just multiply again, 1 over k times this. So we have n factorial in the numerator in the bottom, then we have a k, k minus 1 factorial times n minus k factorial. That's where the first term comes from. Again, it's just multiplying it through by this guy. That's it. And then the same is true for the other two terms. I've rearranged them slightly just so that you can see where the factorials are going to pop out here in the next step. That's it. That's all it is. Okay. So let me try to at least get one line in here. Oh, yeah. Okay. So I can just scroll down instead of actually having to get a new page. Okay. Yeah. So I just used the sidebar here then. Okay. Great. Thank you. Okay. So this is equal to... Okay. And I want you guys to think about this as we go through this because it's really not, like I said, it's really not that bad. And I'm going to go through it slowly, maybe more slowly than I need to, but look at... I don't want to mess up the notes here, but look at the k times k minus 1 factorial. So what is... Okay. What's k minus 1 factorial? It's 1 times 2 times 3 all the way up to k minus 1. Right? So what happens when I multiply that by k? I get k factorial. And it just comes from the definition, really. But it is. It's just the definition. K factorial. Okay?  And remember... Now, I'm not going to write this down yet, but remember what we're trying to get to. You see the thing we're trying to prove at the top? You see that? That is n choose k. We do have the first term. We just need to get that for the other two. And then we're done. You see that? Okay. So plus... Okay. So n factorial over k minus 1 factorial. Okay. Now look at the last two terms in the denominator here, right here. What do those two terms multiplied together give us? What do we get? Yes. n minus k plus 1 factorial. It's the same reason, right? If some of you are getting confused by the terms, just think about what's happening. Look at this. What's the relationship here? Well, this guy is exactly one more than that guy. That's for sure. Right? And this is all the numbers up... All the product of all the positive integers up to it times the next guy is that guy factorial. That's it. That's all it is. Okay? Once you know you have all the numbers up to a certain point and then multiply together and then you're multiplying by the next positive integer, it's that next positive integer factorial by definition. Okay. And what do we get over here? So what does the numerator become on the right side? And plus 1 factorial, right? So now you can appreciate why I gave you the recursive definition because you sort of need to know that now, right? Okay. So in the bottom, what can I do if I want to combine these two terms together? What do I get? k factorial. Okay. And what is this? n minus k plus 1 factorial. You see that for these two terms here? We just talked about this, right? Yeah, I like this system much better. Okay. I'm just going to change this slightly now. Okay.  You see what I did here? Grouping this, right? All I did is just change the second term of the denominator to n minus the quantity k minus 1. When you distribute the minus, what do you get? You get n minus k plus 1, which is exactly what we started with. I haven't changed anything, right? I'm doing this for a reason you'll see in the last step, and then we'll be done with this. Okay. And so this is, I'm going to do the same thing for the other term. n plus 1 factorial divided by k factorial times n minus the quantity k minus 1, all of that factorial. Oh, sorry, sorry. I'm going to, I'll say what? No, that's, it's correct, but that's not really what I wanted. Whoops, wrong one. Sorry. There we go. Sorry about that. k factorial, and this is just, and really haven't done anything, this is even easier, n plus 1 minus k factorial. You all believe that these two things are the same in the bottom? Right? I just changed the order. n minus k plus 1 is the same thing as n plus 1 minus k.  Okay, so now, again, I think I'll just let you look at your notes for this part, and then we're done now. Remember what the notes said? Okay, do you all believe that? That's n choose k by definition. It's in your notes. n factorial over k factorial times n minus k factorial by definition is n choose k. And here's why I changed the form explicitly so that you can all follow where I'm getting this. Okay, think about this for a second. I want you to think about this. You're going to have to get these ideas down for the homework, so it's good to just think about it right now to see where these are coming from. The second is n choose k minus 1. What's the definition? n choose k minus 1. n factorial over k minus 1 factorial, right? The thing in the bottom, that factorial, times n minus the quantity k minus 1 factorial, right? You guys by that? That's why I wrote it out explicitly so you can see. This just comes right from the definition now. And what about this? Equals, and this is exactly also what we wanted. You may have forgotten what it is we're trying to prove, but this is what we were trying to prove. Think about it. Just take a second to think about it to make sure that you process this. What's n plus 1 choose k? Definition. It's the top guy factorial, n plus 1 factorial, divided by the bottom guy factorial, k factorial, times the top guy minus the bottom guy factorial, n plus 1 minus k factorial. That's exactly what we have. Okay? So there it is. So there's Pascal's rule. Okay, so this is going to be very useful. You're going to use this in the homework a decent amount. Or at least some consequence of this. So I'm going to show you what this is good for. So really what I want to try to do is get through two things. The first is not something that the book explicitly mentions, but I'm going to do this just to give you some added practice. So here's what we're going to work on now. Applications of Pascal's rule. So here's the first corollary, and it says this. For every non-negative integer n, that's an i, by the way. Sorry. Okay, let me just put an asterisk next to this. If 0 is less than or equal to k, which is less than or equal to n, then n choose k is a natural number. There are various ways that you can prove this. I'm doing it this way because I want to show you that how induction works, and this kind of has a slightly different flavor to it. So I want to do this inductively. And this is just going to use Pascal's rule. Now I should mention that it's not immediate that n choose k is an integer. Well, of course it's a whole number. Well, remember how n choose k is defined. n factorial over k factorial times n minus k factorial. Maybe you get five-ninths. You don't, but it's not immediately obvious that you're going to get a whole number that way. And actually, regardless of how you prove this, it's going to require a little bit of work. You can't do it in half a line. You can't. You need a little bit of machinery to do this. So here's how we're going to do it. So we're going to use induction, and this is also going to give me another avenue to explain that there's nothing magical about the number one in terms of starting the induction there. It depends on what it is you're trying to prove. Let me just say this really quickly. Hopefully I don't run out of time, but suppose you want to prove that something's true, not for the positive integers, but say for all the non-negative integers. So in other words, we're adding zero now to our set. The same principle still applies. If you can show that the statement's true for zero, and if you show that it's true for any non-negative number, and then it's true for n plus one, then it's true for all non-negative integers. There's nothing magical about the number one. If you want to prove something's true for all integers and bigger than or equal to negative 4,000, prove that it's true for negative 4,000, prove that if it's true for some integer and bigger than or equal to negative 4,000, then it's true for n plus one, then it's true for everything bigger than or equal to minus 4,000. There's nothing magical. The starting point, and then when you know it's true for something, it's true for the next guy, then all the dominoes fall. That's all you need to know.  Oh, no, it's not a natural number, so I can't use inductive. Yes, you can. It's just that your base step starts at minus 4,000 instead of one. That's it. So that's another reason, again, why I wanted to do this, so that you can see that the scope of induction might be a little bit broader than you may think at first. So what we're going to do is we're going to use induction, but instead of the natural numbers, we're going to use induction on the non-negative integers. Notice, remember, these two steps are not the same. Okay, so how do we do this? Well, it's the same basic two steps as before. The first step is, that should be a one, or an I really, okay. And what we need to establish is that this claims, and what I mean by the claim, I mean the starred claim, is true for n equals zero. So I want you to think about this for a second. Here's this starred claim. When n is equal to zero, what is it that we are claiming? We're claiming that if zero is less than or equal to k, which is less than or equal to zero, then zero choose k as a natural number. That's what we're trying to prove. That's what we get when we replace n with zero. You guys with me on this? It doesn't change. The n changes to zero. So here's what we have to check. And again, I'm writing it out in gory detail here just to make sure that I don't lose anyone. If zero less than or equal to k, which is less than or equal to zero, then zero choose k as a natural number. Okay, remember, just like I said before, the k in the end, these are integers now. K is not 0.2, okay? Everything's an integer. So of course, it doesn't matter, really. But what do you know about k? If zero is less than or equal to k, which is less than or equal to zero, what do you know about k? Well, you don't need to squeeze theorem. K has to be zero. There's nowhere for it to go, right? So, yeah, k is certainly equal to zero. If it wasn't, then zero would be less than zero. If k was positive, then you'd have zero less than k less than or equal to zero, so zero would be less than zero, and it's not possible. Okay, negative zero is the same as zero. Negative zero equals zero, so it doesn't matter. So any such k is necessarily equal to zero. There's only one case to check here. Zero choose k has to be the same thing as zero choose zero. And let's just use the definition again, okay? This is zero factorial over zero factorial times zero minus zero factorial, right? Again, I'm writing it all out just to make sure that everyone sees all the details. You should be able to do this in your head, but, you know, certainly by definition this is zero factorial over zero factorial times zero factorial. And what's zero factorial equal to? What was the definition? One, so this is equal to one, which is certainly a natural number, right? Okay, so I'm going to have to go down now. So let me make sure everybody has this first. Okay, so everybody have this now? Okay, so that's the first part. That was easy enough. The second part is we're going to suppose this claim holds some non-negative integer in. Okay, so what is it that we have to prove? Okay, so we must, I'm just going to write it down, so we must show. Okay, so you have this in your notes, so you put a star next to it. If k is an integer with zero less than or equal to k less than or equal to n plus one, then n plus one choose k is a natural number, right? So we're going to look at some cases, and I'll tell you why we're doing that here in just a second. For now I just want you to follow along here. What if k is zero? Can we prove it if k is zero? Well, we proved that zero to zero is a natural number. We haven't done this yet. n plus one choose k is n plus one choose zero, right? And what's the definition? This should be n plus one factorial over what? What do I want down here? Zero factorial times, right? Okay, do you guys see where I'm getting this? Yeah, no. I'm not saying, I never said that zero is a natural number. No, just that k is between. n choose k is a natural number. The k itself could be zero. I'm not saying that k has to be a natural number. I'm saying n choose k has to be a natural number. All I'm saying about k is that zero is less than or equal to k, which is less than or equal to n. k itself could not be a natural number, but I'm claiming that n choose k is a natural number. But n choose k is one. There's nothing contradicted here. I assure you, nothing's contradicted. n and k are not assumed to be natural numbers. They're assumed to be non-negative integers. And choose k is claimed to be a natural number. Okay, that's a good question. The choose, the binomial coefficient is a natural number. And n and k don't necessarily have to be. Is everybody clear on this point? Okay, so what is this equal to? This is pretty easy, right? What's it equal to? One. So that case was easy. What about this case? Case two, these are sort of the extreme cases. k is equal to n plus one. That's certainly possible, right? The only restriction on k is that it's between zero and n plus one inclusive. So k could be zero and k could be n plus one. Definitely those are possibilities here. Then in this case, n plus one choose k equals n plus one choose n plus one, right? Which is the top guy factorial times the bottom guy factorial times the top guy minus the bottom guy factorial, which is zero because they're both the same, right? You guys with me here? And what is this equal to? Well, it's the same thing as before, right? It's equal to one. Okay, and in the last case, and now you'll see if you're wondering why I chose these, you know, I isolated these. I think you'll see why here. Now, case three. What do we know about k? It's between zero and n plus one. The first case was k equals zero. The second case was k equals n plus one. Suppose that k doesn't satisfy either of those two conditions. Then what do we know about k? Well, it's not zero, it's not the first guy, it's not the last guy. So it's squeezed in the middle somewhere, okay? So then the third condition, and think about it for a second, or the third case, right, if neither of these hold, is that one is less than or equal to k, which is less than or equal to n. Think about it for a second. You believe that? And then it's squeezed between zero and n plus one. It's got to be squeezed between one and n. Okay, so what's the big deal? Why did I do this? Then look back to Pascal's rule. Okay, I'm not, again, I just, I'll let you look in your notes here. I would encourage you to do that. Look at Pascal's rule. Look at the hypothesis of Pascal's rule, and look at what our conclusion was. What was the hypothesis? The hypothesis that was made in Pascal's rule, right? k and n are integers with one less than or equal to k, less than or equal to n, right? That was a hypothesis, okay? That's why I isolated those other two as separate cases, because if those two don't hold, then we do know that one is, that k is sandwiched between one and n. So that does, so now we're in the, so now we've satisfied the antecedent in Pascal's rule. Satisfied, because we know that k is between one and n now. And what's the conclusion? The conclusion is that n plus one choose k is equal to n choose k plus n choose k minus one, right? Okay, now, now you might say, and this is a point I really want to be clear on, and this is another reason why I did this problem. We're going to use the inductive hypothesis now. So we're assuming, okay, now I'm going to, I'll wait until you copy this down, but I'm going to go back up here for a second. You guys have this? We're assuming for n that this asterisked sentence holds for n, okay? What does it say? What does the sentence say? It says that if k is any, I didn't write all this out, but if k is any integer with zero less than or equal to k less than or equal to n, then n choose k as a natural number. That's what this is saying. And we are assuming that for our inductive hypothesis. That is true for n. It's not, and then that doesn't just say one k works. It says any k between zero and n works, okay? That's the thing that you have to remember. We're not using strong induction here.  We have to use strong induction. No, we don't. We do not need strong induction. So what do we have? What do we have over here? Okay, so this is assumed. That's the inductive hypothesis. So we know for sure, because certainly since one is less than or equal to k, which is less than or equal to n, and certainly zero is less than or equal to k, which is less than or equal to n. That's certainly true. So by the inductive hypothesis, n choose k as a natural number, okay? So this guy right here is a natural number. What can we say about k minus one? Here's my question. I'll write it all this down, but I want you to think about it for a second. One is less than or equal to k, which is less than or equal to n. So what can you say about k minus one? Zero is less than or equal to k minus one, which is less than or equal to n minus one, which is less than or equal to n. So my point is that zero is less than or equal to k minus one, which is less than or equal to n. That's certainly true. So just subtract one off everywhere in this inequality. Zero is less than or equal to k minus one, which is less than or equal to n minus one, which is less than or equal to n. So zero is less than or equal to k minus one, which is less than or equal to n. By the inductive hypothesis up here, therefore n choose k minus one is also a natural number. The sum of two natural numbers is natural, therefore n plus one choose k is a natural number. There's the proof. There's still one or two of you, when you think about this, are going to be convinced that this is wrong and you have to use strong induction. No, that is not correct. You should trust my brain, not yours. I'm sorry. You should trust me. When it comes to this, you should trust me on this. So these are natural numbers by the inductive hypothesis. So their sum is also a natural number. So therefore, since it's a sum of two natural numbers, let me not be repetitive here. Oops. OK. That's all right. These wands instead of hands. OK. Done. That's it. We consider all the cases, and we've proved it in every case. Now we're done by the first principle of finite induction. OK. The last thing we're going to talk about, hopefully I'll have time to get this done. I hope I do, is the binomial theorem. OK. So I'm just going to go, I'm going to sort of just change gears because I want to try to get this done. Here's the setup. I'm going to squeeze it in over here. OK. OK. A plus B. Now you're going to notice that I'm not telling you what A and B are. OK. And I'll go into that later. A plus B to the first is A plus B. You all know that. A plus B squared. You learned this a while ago. A squared plus 2AB plus B squared. And you can all do this, I'm sure. You can multiply this out, foil it, what not. In combined like terms, you get A cubed plus 3A squared B plus 3AB squared plus B cubed. And you can actually continue this and you can figure out there is a certain pattern that emerges. Notice that as we go through, I've only done three of them, but A plus B to the first has two terms added together. A plus B squared has three terms added together. A plus B cubed has four terms added together. And so if you write these down, you've probably heard this before, the Pascal's Triangle. Maybe you've heard this. So they just keep increasing by one and you end up with this triangular shape if you sort of center everything. And the coefficients follow a certain pattern. They kind of go up and then they come back down again. I'm not going to go into that because that's not really the point of what we're going to do now. But this Pascal's Triangle business comes from this. It's the fact when you combine all the like terms, you just keep going up by one in terms of number of terms of every successive one. OK, now we're going to look at something called the binomial theorem. And this is a nice formula for computing A plus B to any power. It involves factorials. And I'm going to go through. The proof of this is actually not very long. I'm going to try to get through that. If I don't, we'll finish it up on Tuesday. OK, so I guess I'm going to need another page. OK, so let me go ahead and state this. And I'm going to go ahead and call this. I think I called the previous result corollary one, I think. Didn't I? I think I did. Maybe I didn't. I did? Yes, OK. So this is kind of too big of a theorem to be called a corollary. But really, in some sense, it really is just a corollary of this principle, this Pascal's principle that we saw before. OK, and it just says this. So again, just to be clear, this is called the binomial theorem. Let N be in capital N. In other words, N is a positive integer. Then, first, if you haven't seen this before, this may look a little nasty. But it's not as bad as it looks. A plus B to the N is equal to the sum from 0, K equals 0 to N. N choose K. A to the N minus K times B to the K. Now, just I think you've all seen this at some point in calculus, for example, when you're dealing with series, you certainly saw this sigma sign before. I'm hoping you've all seen this sign before and know what this is. If you don't, it's not very difficult to explain. What this means, this sigma K equals 0, and then N on the top just means that expression you see to the right of the sigma, you plug in 0 for K and you see what you get. You plug in 1 for K, you see what you get. You plug in 2 for K, you see what you get. You plug in everything up to N, see what you get, and you add everything together. That's what that means. So N is fixed. N is not moving in some sense. K is the thing that's moving here. K is ranging from 0 to N. OK, so let's take a look at this. I'll see how far I get on this. If I get caught in the middle of this, well, I may finish it next time, or I may just go and do a couple of problems next time. I'll just have to see how much I have left to do. So what we're going to do is we're just going to use induction. And in case this isn't clear, we're going to use the first principle. We don't actually need strong induction here. So what is the first thing that we have to prove? So there are two things that we need to show. We must, the base case, we must prove this claim, star, when N equals 1. OK, that's the base part now. And remember, now we're proving something for all natural numbers. So we're trying to prove this is true for all natural numbers. 0 is not a natural number. So we're not using including 0 in this case. So we're going to start with N equals 1. OK, so what is the claim when N equals 1? Well, just to save time, I'm not going to write this out. Let's just do it. It's not very hard. When N equals 1, I'm going to start on the right side for now. I think it's easier to just do it this way. On the right side, when N equals 1, it's just the sum from k equals 0 to 1. We're just replacing N with 1 now. And I'm just going to show you that we get the left-hand side in this case. 1 choose k, a to the 1 minus k times b to the k. You all buy that? That's what we get on the right side when N equals 1. And what we need to do is check that it's equal to what? a plus b. That's what we want, because when N is 1, then a plus b to the first is just a plus b. OK, so let's see what we get. All we have to do is plug in two values of k here. Remember what I was saying before? Whatever k equals, that's where we start. And then we keep plugging in values of k until we get to this top number. And the top number is 1. So it's just k equals 0 and k equals 1. We're just plugging those two numbers in for k. Adding them up, we should get a plus b. OK, so what happens when we plug in 0? So this is 1 choose 0, a to the 1 minus 0 times b to the 0. OK, now might be a good time to say something about a and b. a and b can be, they don't have to be natural numbers. a and b can be real numbers. They can actually be even more than that. They could be polynomials. In general, if any of you have taken abstract algebra, a and b can just be any elements of a commutative ring. That's it. If you don't know what I'm talking about, it doesn't matter. This theorem holds in general in a very more general setting than what you're used to if you haven't taken abstract algebra. That's why I didn't say what a and b were. They don't have to be numbers. They don't have to be polynomials. They could be all sorts of weird things. OK, so what do we get? OK, I'll probably go through this a little more quickly than I did before. What's 1 choose 0? 1. What's a to the 1 minus 0? What's b to the 0? So you buy that? OK. Sorry. I didn't let me put this parenthetically now. Equals a. This isn't actually what this is equal to. It's this plus the next guy, right? OK, which is just we're plugging in now 1 for k, right? 1 choose 1. a to the 1 minus 1 times b to the first. And what's this equal to? Right. And then so we get a plus b, which is a plus b to the first, and which is what we wanted, right? So the theorem is definitely true when n equals 1. We just verified that. Second part. And this is a little trickier. It's just going to use Pascal's rule again, though. We're going to assume for some n and n, some little n and big n. And again, I'm just going to use the same notation as I used before. We'll just put an asterisk next to this. a plus b to the n is equal to the sum from k equals 0 to n. And choose k a to the n minus k times b to the k. So because I'm going to have to go down another page anyways, I'm not going to write out exactly what we have to prove. If you want to put this in your notes, you can. I mean, I'm going to prove it, but I'm just not going to say it beforehand. I think I'm going to run out of time here. But what's the statement when we replace n with n plus 1? It's a plus b to the n plus 1 equals the sum from k equals 0 to n plus 1, n plus 1, choose k, a to the n plus 1 minus k times b to the k, right? OK, so that's what it is that we have to prove. And we can assume that this is the inductive hypothesis. We assume that we have it for n. And we're going to use that fact to get it for n plus 1. OK, here is what I am going to do, because this part is just a little bit technical, and I don't want to get stuck in the middle of this, actually. I'd rather just do this in one class period, because otherwise I'm going to have to do it again anyways to refresh your memory on Tuesday. So I'm going to just stop, I think, at this point. What I'm going to do is I'm going to give you a homework assignment for this section. Again, Tuesday I'll finish this up. I will definitely do a couple of problems, again, like I did last Tuesday, to get you up to speed. I'd like another week or two to flip the assignments due date back to Tuesday, so that you have a weekend to kind of work on things. And you've got a little bit more time than just two days after I do a couple of examples. But we're going to try to get there soon.  Sorry I didn't quite get through the rest of this, but we'll finish this up on Tuesday. OK, so this is to be continued. Although I will say this, OK, the homework that I'm giving you, I would say the majority of the problem, the book actually gives you hints to these. And the book also works out a couple of examples. So really, you, I think, have the tools you need to be able to work on the homework already. Some of this is going to involve induction again. But you're just going to assume the binomial theorem is true, and then just kind of work through the homework, just pretending that you know what's going on. OK, so here's the homework. 3A, 3C, 4C, 5A, 5B, and 5C. Could have given you a lot more than this. But that is all that you have to turn in for next Thursday. And if anyone, I don't think anyone came in that late, but if anyone didn't turn their homework in, definitely get that in before you leave. So I think that's all I wanted to say. So all right, we'll have a good one.


So, hi, I will put on Spanish there, which is an alternative hardware description language for digital hardware So during this presentation, I will first give you a little introduction about Spanish there how it come how we can you how we can use it and Then I will give some different with the actual and very long and Then I will give you many little example to show you the inventory of Spanish del and how it is different What it gave you by using some abstraction and By using some software engineering approach to do hardware design So this presentation will okay this presentation will only be about synthesizable hardware. It's really not about Simulation thing You can simulate the output netlist so Yeah to simulate generated the hardware you can use Regular tools because the output netlist of Spinal is the actual and very long so you can do the simulation of tools output netlist So, yeah, this is the context. I don't have many time to to give you my opinion on that But basically what I think is the actual 2002 and very long 2005 are really bottleneck in term of engineering by many aspects and The actual 2008 and system very luck will not save all of us because of idea support because even of features that they bring That are not so good. Finally, so I don't really have many times talk about that. Sorry, but So Spanish there it is an open source project which starts in December 2014 It focus exclusively on RTL description and as you can see here there is a flow so You can describe your hardware by using some color files Then you ask it's a Spanish deal compiler to translate it into a regular visual and very luck that you can synthesize or simulate and so this made Spaniard directly compatible with all idea tools because it's regular vehicle and very dog and You can also integrate some legacy IP like a block ram for from tidings or from matter I don't know are some more complicated things Inside the Spaniard there Spaniard here keep using a blackbacking system. So it's quite compatible with existing flow and existing IPs Then about the abstraction level is it started the same level than VHDL and very long a Little bit different But the thing is then on the top of that you can build a new abstraction level You can go away from this mess of fire which is hardware design to really say what you want So yeah just before example some last points So there is no logic of a head in the granted code because by next day is not a each HLS approach It's really much more like an RTL approach as the idea in very long where you have to design gates Why you have to design registers and six like that. So there is no overhead. You will not lose performance by using it all names and all Your component hierarchy is preserved during the translation from Spain LXL to very long and VHDL so you can Simulate it by using your favorite simulator. You can look at the wave and then very easily Understand by understanding the wave catch up where is the bug in Spain LXL if you have one and Basically Spain LXL is not ready a language and this is the main point about Spain LXL Spain LXL is a language It's not is a scalar library So a scalar is a gonna help purpose programming language and Spain LXL is a library implemented on the top of it to allow you to describe being your hardware and It could seem strange to have a language into another one But it is probably one of the one of the best point of Spain LXL. So I will comment that later So yeah, this is a simple example a very random one which has no interest in practically But imagine you want to design this little piece of hardware with a single condition to drive some combinator logic Some register without reset some register with reset in VHDL and in very low because you have to use a simulation constrict to build hardware to infer the hardware because VHDL and VHDL were initially made for simulation purpose not for inferring hardware So you will have to write three different process You have a lot of redundancies because for example this condition is depicted three times Which is not very clean which is not very Safe because you can create bugs by having duplications at many place So this is how our user to do and This is how we can do with an hardware dedicated syntax and this is some Spain LXL example So first difference you can define things explicitly as signals like my register Is a register not because you are assigning it into a clocked process But because you define it directly at the definition. I want a reg of this data type I want a reg of the data type with a reset value to zero So because things are explicit now you you don't have to write process and these kind of things and You can write for example that that you want my signals to be false by default And then when the code is true, then you can assign all of those guys in the same conditional statement So Another little example This one is to compare and just to understand the abstraction level of Spain LXL right start So if you want to define a simple timer component as you can do in VXL and Verilog It's very similar like you can define. Yeah It's based on Scala. So it's much closer though to Software design in terms of syntax can define a timer class with some construction parameters This is like a generic or a parameter in Verilog Then this class with the extend component to say this class is a component So it's a yeshical element of my design then can define IO into input-outputs into a bundle bundle is the concept of hardware data structure in Spain LXL Then can define inputs and outputs of your components. It's very easy Then can define a counter into insane your timer, which is a register of this data type Then you can write conditions assign it all the conditions Reassign it so last assignment to it like last assignment to it like in VXL and Verilog then you can do assignment so If you know how to do VXL and Verilog, it's not so far. You can use it by the same way It's usually not have to care about process and things like that but then I will start with the fun stuff so imagine you want to Have a n-shake bus of color So by a n-shake bus of color I mean a stream of data which carry colors like a valid and already flag as arbitration and a payload as an LVB data structure and Okay, you have a source n-shake bus of color You want to queue it inside the 5.0 and then you want to connect it to the sink n-shake bus of color So in VXL 2002 it is really kind of boring because you have to define X signals one by one In many cases you can not really use records there because you want to have them Pametrized by Verix, so you can't really use records there in all cases Then the most boring part is probably about instantiating the component itself where you have to bend each signal one by one Which is very error-prone, which is very copy-paste work, which is a donkey work, maybe so In Spanish there things are much more object-oriented, which means So yeah, for example, if you want In Spanish there the concept of handshake bus is named stream So if you want to define a source stream and a sink stream you say I Want a source and a sink which are stream of RGB color with this parameterization So you can already much more use that as stricter which which have parameterization Which have inner parameterization you don't have you are not bound at this level and As instance to insert this FIFO you just say I want a FIFO, a stream FIFO with this data type has Element which are stored by the FIFO you don't have to bend all signals of your data structure into a single SD-logic vector anymore Then you say the depth that you want and then this is the thing if you want to access as a push port of the FIFO You say FIFO.io.push. You don't have to bend things. You directly have access to them By using this kind of object oriented way like a FIFO is an object You can say I want to access this attribute of the FIFO Then here for example, it's written that you take the source Stream then you connect it to the push port of the FIFO and here it takes a pop pot of the FIFO And you connect it to the sink so it is better than this Already, but it could be better and I will come and a better way to do it. So basically stream Is not something that come magically from spain-axial compiler It's something which is implemented on the top of it by using regular syntax of it of spain-axial So it is a class with a type parameter here Which extend bundle because this class represents a hardware data structure so it should extend bundle Then you can define element into your data structure has a valid flag and ready flag a payload instance of this parameter visible type so you can use this data structure has Slave input of your component or as master output of your component It's not like records where you have all signals that can only be in one direction It's not explained there how to do that, but it is really useful and easy to do and Then there is a thing these data structure you can add functions in functions in it like the operator that we have seen there to connect In in very long Function tasks and procedure are not that much useful because inside them you can't define what you want you have to define relationship between outputs and inputs of the function, but you can define inside them Register you can insorciate a component you can do all the tricky stuff that you want You are really limited with them But here you can really do what you want like you can define a queue function We take a parameter how many big you want the queue to be and then this function will create for example a new FIFO internally a new component so and then it will connect The stream on the one who you are calling it to the path part of this FIFO and This function will return the output part of this FIFO. So it is it is really simple But in fact there you go in place of writing all this stuff which is already better Than regular VHDL. You are just writing this like you want a source and a sync stream of RGB colors And then you'd say you take your source you want to queue it We have FIFO of 16 element and then you connect it to the sync and there you are starting to going away of the mess of wire of Hardware design you are just saying what you want and this is working You can you can write a bug like in that you can do mistake in wiring. You can do these kind of things So it's it's not only you can go further than Like imagine you want to do this hardware like you have a Source stream of color. This is our random example, but it is an example Imagine you have source Handshake source stream of color you want to draw a transaction of this stream when they are black So you need to add this arbitration gates and this check Then from this point you want to add a pipelining stage to have better maximum frequency or to have a one Depth FIFO storage. I don't know So you need to add that So in real there Probably if you have to do that you will do it by hand and it's really error problems If you do error in the arbitration, it's always a lot of time to find out where is a bug in your arbitration, but here There is how you can do it in spain agile I mean how you can do it by using the library which is implemented on the top of spain agile Like you can define a source stream of FGB color. Then you can say I take my source stream I want to draw a transaction when the source payload is black and then which this function through and we'll return this stage This this point and then you say I want to stage it and this stage function will create this hardware for you And then it will turn this this node and then you name it sync So really you are you are you are saying what you want and not how really it work Signals by signals and this is not magic here. You can do You can navigate this to the implementation of this function and this is pure RTL implementation It's not from the spain agile compiler so another example state machine so You can design sit machine as you do in vex then and very log by using switch statements and things like that But you can also use the state machine tool so sit machine tool is one time again implemented on the top of spain agile It's not something integrated in it and If you want to do to state machine So you can say I want a new state machine. I want state ABC, which are states state air is my entry point then you can define some signals inside it and then You can say state be an entry. I want to set my counter to zero State be when it is active I want to come top when my counter is for I want to go to state C and then on state B When it exits, I want to put my IO result to true So it's just an example that Spanish they'll allow you to rise your level abstraction But it's not because it's implemented in it. It's because its syntax is flexible enough to allowing you to build this abstraction state machine is not magic really is just software engineering with How do I design combined together? So another example imagine you want to do that you have A bus like an axi lights on a PB one I don't know then you want from this bus be able to drive a signal A and B which are inside of 32 bits So you want to have some register here which are right only from this bus Then you do some calculation like multiplication between both and then you want to be able to read the results from this bus so if you want to do that in the engine very long you will have to Yeah, to manage the bus timings by hand to do some switch cases So it could be a right, but it's not the only solutions So here is an example for example if you want to create an axi light for this you can write it like that And then you can use an axi light slave factory tool So this tool is one time again content of the top of spin axi It's not integrated inside it and this tool will allow you to specify your register mapping by an app by an abstract way without having any any knowledge of the boost timing then Like you can say a is Something created from the factory has a right only register of this data type mapped at this address and the same for B and Then you can do some calculation here with results and then you can say I want my factory to make the result readable at the address 8 and So this specification here is completely abstract from the fact that it is an axi light one You can replace it here by a PB one and it will work directly so and behind that there is a Lot of software entering like there is Ash map that you can use to elaborate your design and this is used behind the center there is Abstract classes there is abstract functions. There is a written see heritancy. Yeah Inheritance, exactly and yes, there is many things that you can do So a last example This is just a demonstration project that I have made fully in Spanish there just to check that it's work correctly And it's not too much buggy and so this is a kind of little SOC Which work on FPGA with an RISC 5 CPU with instruction cache, a GTAC chain to debug it Some SRM controller, one chip RAM, a PB bridge for low-performance peripherals, GPIO, timer, UART and VG This is an axi For interconnect is in a PB interconnect and there is some sample of code which are interesting inside it Like for example to implement to sample to instantiate this a PB bridge from axi light from axi 4 It's just writing that you say I want my bridge with this parameter section and this is done and to instantiate all this Decoding stuff for the a PB with all those connections between components is just saying I want an a PB3 decoder My master is the a PB bridge IO a PB and there you can give a list of save like Timer controller IO a PB is mapped at this address with this address range and you know, it's Invigil it's and in very luck is so boring to do that. So Here it is Saying what you want and you get it Then there is another pattern though. It is for the axi 4 side and this one use a data model Like you can create an axi for crossbar factory. Then you can feed it with some data model by saying I have this list of slaves I'm up there. You have this lift this list of master and They can access those slave and then you say I tell you everything. Please build yourself so you can really do software engineering to design your hardware with this approach and Yeah, this is this is all for this presentation about Spanish deal You can find everything open source and teeth repository. There is out like the commitment which start to be Good, I think there is some ready to use project to help you if you want to trade and some communication channels Or do not hesitate to To comment them if you have some issue to start up with tools or anything so This is all thank you Sorry To compare it with chasel so basically I start to work with chasel I was happy some months with it. That was two years ago and or three years ago but basically It appeared that there was a lot of issue with it and like for example clock domain support which has very Badly done and there has regularly some issue about it, but no No, but there was no move on the chasel stuff so You can also find a list of things that I don't agree with chasel in the design on the fact section of the online documentation and Yeah Maybe chasel is much more about a very large implementation of the idea Spain actually is much more about a VHD and implementation of the idea which is with check more things and most of But really it look like in surface is the same but really there is many difference behind the same like Yeah, you have to check the fact So So basically Yeah How to map some pure scala into hardware? It's your question like using hash map and things like that No So you If you can writing Software and make it able to run in hardware and software. No, it's not the case So Spanish Dell is not an actualist approach It's really an ethical approach where you use scala as an elaboration tool to design what you want by saying I want this register I want this register I want to do this operation but Spanish is not about translating and algorithmus into Hardware it's much more about saying what you want Explicitly and you get it but you can use hash map to do the elaboration of your hardware You can use all these kind of things like dynamic list and but has an elaboration tool like a for generating vex. Del or if generate in vex. Del This kind of things Yeah, so how I specify a clock for flip-flop though, but it really it's much better than vex. Del because It is implicit so you can define area of your design Which are under the effect of a given clock and reset configuration and then everything inside it all Subcomponents will automatically get it So, yeah, I had a slide but I haven't the time to talk about it So which are a message you get when it doesn't work so basically you have much more confidence when you generate the Spanish deal Netlist you have then much more confident than it's working physically in for real than a handwritten vx. Del one because Spanish deal do a lot more check by default than I think yeah, it do a lot of check directly than Like for example, it's like that you don't have comment wall loops. You will check that you don't have with mismatch these kind of things and If you have an error it will print you where in your Spanish. Del description. There is something wrong It will give you the name of the signal. So really it's made me even less easy than vex. Del to remove bugs from a Spanish their description because Yeah, I made though it check a lot of much more things and I put a lot of intention to have nice printed error message with the print stack of the execution of spain a little description And these kind of things so it's not really messy it Right have to try to give and I could get a good idea of it Put in the discussion session So there will be three process granted from it exactly like that exactly the same Oh It's better like they're all is How did this thing is in between blocking and not blocking so if they're in Spain like they're by default it is I don't know which is one, but it is like the default Well, like this one here. Yeah, you can do the other one by using another syntax, too So how the multiplication is translated into vex. Del exactly the same than in vex. Del so Then if you want to use SPGA the ticketed blocks and things like that You can associate a black box or you can try to Write it by the way that's the synthesis tool we understand it But there is no magic between in the translation so it's one to one So if you to about formal revision between the input spin ideal and Output there is no formal verification about it because basically it's not required because you are verifying the output netlist You are not doing your verification stuff on the spin ideal description. You are verifying the vex. Del generated Five so it's why I don't really care about it being formal, but I have a good confidence in the translation process


YouTube team Wow Team keep it clean. What is going on? It's engraving here with another video and We're real quick Ravens fans. It's over. So I mean, I know a lot of Ravens fans They didn't think we would really get them But there was just a little shimmer of hope somebody that could possibly take over for two real sucks and really hold it down as a Past Russia slash DN slash run stuffer slash all that good stuff But it's over now. But anyway, it really never got started. So in a big trade the chiefs and the Seahawks The the Seahawks they put it out there. I didn't think it was gonna go down. I really didn't I Did not I thought they were gonna trade them I didn't think they were gonna pay him, but I did not think that they would actually get a first round I didn't I Didn't even when you look at the numbers the numbers are great numbers don't tell the whole story, but The numbers are great. You look at two thousand six last three seasons 2016 He had 10 sex. It's pretty good 2017. He had nine sex. That's pretty good 2018. He said man, you know what? I'm trying to get paid. Let me up my game even more. So he got 13 sex and he got picked to And he got picked to he just knew he knew he's getting ready to cash in So the the Seahawks and the chiefs and there were some rumors floating around about Patrick Peterson possibly being traded But we see that that that died down. But anyway the Seahawks and the chiefs So the Seahawks send Frank Clark and a 2019 third-round pick to Kansas City Okay They they sent him and they sent a third-round pick really all gave up a third-round pick to get rid of him But what did y'all get? What did you possibly get in return? Well? When you see what they got in return, you can understand why they were willing to part ways with a 2019 third-round pick so they got the chiefs First-Rounder the chiefs 2019 first-round pick They got a 2020 second-round pick And they got the chiefs 2019 third-round pick. So they they came up Wow, that is crazy, man That's crazy So the Seahawks got two first-round picks in this draft And the Seahawks they've been a pretty good team at drafting and developing players and The chiefs the chiefs I guess they felt like you know what we're willing to give up a first-round pick because we feel like Frank Clark Is a sure thing. He's only been in the league since 2015. So he's super young and that is Some some some fans of teams don't like when their team gives up a first-round draft pick Because it's like man, you can get a young guy coming out the draft And that's true and you can understand why but at the same time, it's like mm-hmm when you If with the draft and I mean with enough players to but more so with the draft is a correction It's a gamble. It's a risk. You never know how somebody's gonna pan out And not to say that you shouldn't take those risks, but it is a risk No matter how good or how great they were in college. It doesn't always translate to the NFL But now if you got a player that's been in the NFL you have NFL tape on him You've seen him against the top of the line players in the world Since he's he plays for the best league in the world. He plays for the highest This is the highest that you can get at football. This is the highest level of football that you can play So you've seen him going up against the toughest of the tough competition in the big leagues And you know what he can do You know what he can produce giving up a first-round pick is not so risky We saw it last year with the Cowboys with the Marnie Cooper We saw it last year with the Bears with Khalil Matt Teams are willing to give up first-round draft picks for guys who they feel like Can come in and play right away. They know that stuff. They know what they can do They know that they can get it done so While this at the first time at first it was a little surprising When you think about it It's not really surprising that they were willing to give up a first-round pick Well, they were the first round pick and a third round pick from this year so that's That's a pretty hefty price. That's a pretty big-time draft capital to give up but they said hey, we're getting a short thing and I absolutely love how the Chiefs are taking care of their defense. They weren't gonna pay D4 It's kind of weird that they didn't want to pay D4 But they're gonna get ready to pay Frank Clark That's one part that I'm kind of like scratching my head about a little bit I'm like, maybe they feel like Frank Clark won't line up off-sides as much. I don't know. But anyway, um The Chiefs are getting that guy and I love how they've been taking steps They've been taking steps to upgrade their defense because defense was their major Downfall this past season. It was defense Chiefs then put up and I think the lowest amount of points that they scored in the game was like 27 I think I believe it was 27 because I know I think they scored 27 against the Ravens And I think they scored 27 against like the Broncos and that was a low for them I believe don't quote me on that. I know it's your chief saying to let me know they'll correct me They always do but anyway this uh Cheese man, like I said, they taking some steps like honey badger Honey badgers another one they went in and got honey badger and They are trying to make sure like hey, we we ain't messing around We know that our defense sucked last year. Um, they the past rush was nice But the overall defense not especially on the back end especially on the back end Past rush was the best thing about their defense last year But that was literally the only thing about their defense last year now the cornerback that they that they lost Who went to the Steelers? He kind of um How well he played always forget his name as a Stevenson always forget his name But how well he played sort of got lost in the overall just bad defense of Kansas City So again, they've been taking these these steps To try to fix that which I really really really do like that's why I put on this this this team keep it clean hoodie There's red one I'll say that you know what chief did a trade. Let me put on the team. Keep it clean the red chief's colored hoodie You can get yours at the team keep it clean stores down in the description and it's right out of the video too Sad to throw that in there, but anyway and the Seahawks Seahawks they get prepared for both this year They get some extra preparation for both this year and next year to just get some young guys a lot of people figure when Russell Wilson got His deal when Russell Wilson got paid when he cashed out Did he make a little video in bed with Sierra and then he somebody said um somebody said man He got his bag. So now he talking like he really want to talk because when they went to the day they showed the video of her braiding his hair and He wasn't talking like oh, I'm Russell Wilson. He was talking like. Yeah, I'm Russell Wilson Um, but somebody said when he yeah when he got it when he got paid then the real Russell Wilson came out But it's all good. Anyway, um Lot of people figure when he got his money from the Seahawks when he got his contract a Lot of people just saw the writing on the wall They were thinking like the Seahawks are not about to pay both Russell Wilson and Frank Clark they're not gonna pay both of them and What happens they don't pay both of them and they get rid of Frank Clark And it was around the time when Russell Wilson first got his deal Right a little bit after that. That's when all these trade rumors started starting about Frank Clark Where's he gonna go with the Seahawks? Oh, they put them on a trade block Seahawks within the trade Frank Clark So what happens? Boom now they trade Frank Clark. So and he's getting ready to get paid too The chiefs are gonna pay him. So again, I'm still I'm still a little confused on Why why not default? Because it's that's just that's the only only part of this trade that I question and they gave up a lot too They gave up a lot chiefs. Um, I'm sure chiefs had multiple offers, but when I Mean, I'm sure the Seahawks had multiple offers But when the Seahawks saw what the chiefs were willing to give up they were like, oh Gosh like y'all are really giving us all of that And they were like, you know what they probably felt bad they probably felt bad They were like these dudes are really giving us a first and a third round pick this year and the second round pick next year They said hold up. Hold up. Hold up. They had a check They had to check that the draft bank account see all the picks that they got and they said oh, you know what? We we can't go out like that.  Let's give them a third round pick. We feel better now Let's give them a third round pick because we don't want this image of us to to be portrayed as as bullies so they gave them that third round pick as type of compensation But it's official now It's it. Well, it'll be official soon Because Frank Clark, uh, he wants to he wants to get paid and she's gonna pay him So she's a team now Ever since I mean last year with how well that offense played and if they have Tyreek Hill this year We'll see what happens with him in the whole all that situation Because we know recently they said that they lost him and his girl him whatever she is to him The mother of the child they lost custody of the bike of the baby Said they removed the baby from the home So NFL could take that and with the whole personal conduct policy, they could be like, oh wait a minute They remove your child from your home some stuff that was going on. You know what suspended three games Suspended forget we'll see we'll see So that will definitely be Definitely gonna be keeping an eye out on that and keeping an ear open for any news we hear on Because that will play a huge role With the chiefs with their schedule Just with everything and watch watch with the draft to Watch pay pay close attention to the chiefs draft reason I say that is because Pay attention to that draft to see if They draft a player who may be similar to Tyreek Hill Reason I say that is because he may be missing in action for a little bit and Chiefs may feel like hey depending on how this situation goes with him And I'm not trying to put anything in your head, but just something to think about Depending on how the situation goes with him with all this stuff off the field The chiefs may be looking like you know what let's let's find somebody who's similar to Tyreek here Because they may be hesitant to pay him cuz he's up. He can be keeping get paid now He can get paid now And you know, he's one of the better wide receivers in the league Consistent he got a hand he's small, but the dude is dirt fast and he's a he got some really good hands And he catches stuff on contact too. Hey, I'm a Ravens fan We we saw this dude on fourth and what fourth and 19 and on a fourth and nine in the same game We saw poor just sad memories that game hurt my heart We saw this guy catch the football and get hit right in the back held on to the football Stayed up and then ran for like 10 more yards Granted it was against Eric Weddle, but He still did but anyway, we'll see how everything goes down So just like Frank Clark with the Seahawks I'm out


All right today, we're discussing the National Park Service as you may know I'm actually doing the same thing that I'm teaching you guys out there I'm working with the federal government and I'm trying to solicit them for business. I'm looking for work I'm looking for projects and I have a client of mine that we're working with up in the Northeast United States and One of the potential clients that we're looking at one of the potential agencies I'm looking at is the National Park Service. So I had the privilege of speaking with someone from the Denver Service Center the other day I spoke with a person from National Parks and they gave me some good ideas And I want to share that with you and all of you out there so that maybe this could potentially help someone That's watching my YouTube channel. That's following me landed contract with the National Park Service. All right, so let's take a look at the National Park Service and what I want to show you right now I'm over here on my computer and as you can see If we type in National Park falls under the Department of Interior. So let's type in Department of Interior Forecast list we click here click forecast and it lists the procurement opportunities fiscal year 2018 forecast list Now the information that the young woman share with me Here's forecast list if you click this link It pulls up an Excel spreadsheet I'm gonna go ahead and pull up that spreadsheet now and all this spreadsheet you will see is a list And that's what it looks like Actually, it's letting me write the bottom of the page but on the spreadsheet, which I'll find there's 1951 opportunities and they're organized by contract types. It gives you a name of the opportunity It also tells you the value range Of the job it tells you where the place performance is an appointed contact name and number All right, and this again this is for 2018 projects So right now We are in july 2018 the fiscal federal year ends in september 30th of 2018 Therefore you have about two months or a month and a half left before the fiscal year ends The information that they suddenly provided to me was that a lot of times Even though the Denver service center is the place or the distribution point of all contracts for national parks The local parks also they give out their own procurement And for a lot of people out there that are looking for a single simplified acquisition or micro purchases This is a great opportunity for you to solicit directly to a national park in your local area And also find a potential opportunity or maybe find an opportunity to work with them. So what happens is I understand from my phone calls weekly phone calls with subscribers like yourself that a lot of people are wanting to small off with Small opportunities under the $10,000 threshold under the $25,000 threshold This is your place to go to do that and also be close to your Geographical location whatever city-state that you live in so it's it's I want to compare it to More like a state or a local contract because it's close in terms of proximity to your actual physical location However, you're getting federal past performance federal jobs and it's also a small project So if you want to if you're afraid of going up to the larger things or if you don't have past performance Um sufficient past performance you believe to pursue a much larger opportunity This is the place that I would say that you could go right away You there's probably in your local area. They may be 10 or 15 numbers So it's very easy manageable a number of people to call to reach out to to solicit Nothing extraordinary, but at the same time You gain the benefit of working in the federal marketplace. So as you can see There is 1950 opportunities to have them in all states. Everyone has a national park in their state You could sort by states and you could pick or choose the ones that are closest to you So this is a way of using the forecast list to identify potential future clients for jobs and opportunities coming up Even if you're not successful in landing something right now You still have the opportunity you're still building relationships You're still making connections and from my understanding if the person at the local level, right? They like you and they want to do business with you They can then refer that back up to the Denver service center for larger contracts for larger opportunities for set-aside jobs for Negotiate type proposals. All right. So this is your tip it for today. Take advantage The national park service their forecast list of almost 2000 opportunities for the last 60 days left before the federal fiscal year ends