Problem list

How many rearrangements of the letters "RADIOHEAD" are there such that the D's are separated by at most two letters? (e.g. ADADIOHRE is such a rearrangement but RHDEAADIO is not because the D's are separated by 3 letters)

Answer: 52920

Obviously the question here is not to calculate the number of such arrangements since I have done that for you. I would like you to explain your answer using the following three tools at your disposal (all from Chapter 3-1 in the book). You may assume these have been given to you, everything else needs to be explained:

1) The addition and multiplication principles
2) C(n,r) = the number of ways of choosing r elements from a set n distinct elements =n!/r!/(n-r)!
3) P(n,r) = the number of ways of ordering r elements from a list of n distinct elements = n!/(n-r)!

I will use the same criterion that I use to evaluate the forum questions, namely I am looking for clear, complete and short explanations. The 'short' criterion is is important so please do not have a solution which goes on for more than a page (because I won't be able to follow it and then it breaks the 'clear' part too).


When you write anything using the words that do not clearly have anything to do with the problem like 'choose' 'list' 'spaces' please make it clear what you mean. Somehow you must explain how you transition from 'number of rearrangements' to an actual value and so you must make it clear to someone who starts reading a question about rearranging letters how you are applying the tools you have available to you to arrive at a number value.

Back to the web page for Math 5020.