How many distinct six-letter rearrangements of the word ``Hawaii" can I form?
I notice that the word ``Hawaii" has two repeated letters: A and I. This means that not all six-letter arrangements of the letters H, A, W, A, I, I will be distinct. For example, swapping the two A's or the two I's will not create a new word. So I need to account for the fact that some arrangements are identical because of the repeated letters. One way to do this is to use the formula for permutations of n objects, where r of them are identical of one type, s of them are identical of another type, and so on. The formula is n! / (r! s! ...), where ! means factorial. In this case, n is 6, since there are six letters in total. r is 2, since there are two A's that are identical. s is 2, since there are two I's that are identical. There are no other repeated letters, so I don't need any more factors in the denominator. Therefore, the number of distinct six-letter rearrangements of the word ``Hawaii" is 6! / (2! 2!) = 180. 