A. For 4 coins, count explicitly how many different ways there are to get each of the following macrostates:
4H, 0T
3H, 1T
2H, 2T
1H, 3T
0H, 4T.

B. Now suppose that you had N coins. Create an mathematical expression that would allow you to calculate how many different ways you could create a string of flips that would give M heads and (N-M) tails. Consider a set of N coins that have M heads and N-M tails showing. How many different ways could you choose a sequence of the coins? (Hint: You could choose the first one in N different ways. You could then choose one of the remaining N-1 in N-1 different ways; etc.) Since we don't care what order we get the heads or tails in, you have to divided by the number of ways of permuting the heads and the tails. This result is called NCM, the number of ways of choosing M objects out of a set of N without respect to order. (What you are to do for this part of the problem is justify the expression for the number of combinations in terms of the relevant factorials by describing the choosing and arranging process.)


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