Mississippi

The word "Mississippi" has one "m", four "i"s, four "s"s and two "p"s. Since the position of two of the "s"s has always been determined, we only need to find the number of permutations when we have one m, four i's, two s's and two p's. Since there are 9 letters in total, there is 9 possibilities for the first letter, 8 possibilities for the second letter... which give us 9 8 7... all the way to 1. That means that there are 9! permutations for rearranging the letters. However, note that we have more than one "i, s and p". Miissippi is one possible permutation. If we were to switch the position of the p's, though, the arrangement of letters would not look any different than it does now as all p's are the same. However, when we calculated 9!, we counted both permutations separately. Therefore, we must eliminate some of these possibilities. There are 4! ways in which the i's could be rearranged among themselves but not give different permutations, like in my example. (In a permutation like iiiimsspp, there are four i's that could be in the first position where there is an "i", three i's for the second position... or 4 3 2*1.)

Similarly, there are 2! ways that the s's and 2! ways that the p's could be rearranged among themselves but not give different permutations. Divide 9! by 4! 2! 2! to receive 3780.

Answer is 9! / (4! X 2! X 2!)

I didn't look at the category, assumed this was a computer science problem, and dove right in. Here is my solution in Python:

import itertools
print sum(1 for x in set(itertools.permutations('mississippi')) if x[0]==x[-1]=='s')

Obviously, the pure math solutions are better, but I'm showing my code here for education's sake. For example, I did not know that the permutations method did not automatically get rid of duplicates, hence why I had to turn the data into a set. You live, you learn.

9! / (2! * 2! * 4!) = 3780