How many anagrams of a word exist

Question

How many anagrams of the word ''Combinatorics'' exist such that a consonant is in the middle.

Answer 1

I shall assume one does not care about capitalization.

We then have these letters:

2 C 2 O 1 M 1 B 2 I 1 N 1 A 1 T 1 S

The total number of permutations is $\frac{12!}{2!2!2!}$

The number of words with a vowel in the middle is the number of words with an O in the middle plus the number of words with an A in the middle plus the ones with an I in the middle. How many words are there with an A in the middle? $\frac{11!}{2!2!2!}$. How many words have an O in the middle? $\frac{11!}{2!2!}$. With a similar train of thought, we get that there are $\frac{11!}{2!2!}$ words with an I in the middle.

Then the number of words with a consonant in the middle is

$$\frac{12!}{2!2!2!} - \frac{11!}{2!2!2!} - 2\frac{11!}{2!2!}$$