combinatorics – Why do we divide by the count of each letter in questions of the form “how many words can I form from the letters in”?


For example, how many words can we form from the letters in the word google?

First I thought you counted how many different letters there are in this case 4, therefore in each spot (6 spots) there are 4 different choices. So the amount of words is $4^6$?

I found out this is wrong and instead you use the idea of a multinomial and calculate $frac{n!}{a_1!a_2!,…,a_k!}$ where $n$ is the amount of letters in the word, here n = 6, and then $a_1=”g”$ which appears twice and so forth.

Why do we divide by the $a_k!$ factorial terms? Do we not lose possible words?