# [Math] permutations with the English alphabet

combinatoricspermutations

How many four-letter words, using the English alphabet, are possible if letters if only vowels may be repeated? How many four-letter words are there if at most one repetition of any letter is allowed? Or if exactly one repetition is allowed?

I am trying to answer these questions. I know there are 26 letters in the english alphabet. if no letters can be repeated, it would be 26*25*24*23. however, my problem is if only one repetition is allowed, i don't know which letter would be repeated. 26*26*25*24 is different than 26*25*25*24. how do i get around this? I was wondering if there would be a way to subtract from the total possible number of words(26^4). Or add the four letter words with no repetitions (26*25*24*23) plus the ones with one repetition. Unfortunately, it seems like im still stuck with that one-repetition part. any help would be greatly appreciated!

If we have exactly one repetition is allowed, we start by choosing our three letters: $\binom{26}{3}$. We then choose a letter to be repeated in $\binom{3}{1} = 3$ ways. Finally, we permute our letters in $4!/2!$ ways by the multinomial distribution. By rule of product, we multiply, to get:
$$\binom{26}{3} \cdot 3 \cdot 4!/2!$$
This is correct thinking. Notice that the words with no repetition are disjoint from words with exactly one repetition. So by rule of sum, you add $26!/(26-4)!$ with the quantity I noted above.