Competition problem — Combinatorics

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Combinatorics Difficulty 5.1 AIME, harder Find the answer

Call a string of letters $S$ an almost palindrome if $S$ and the reverse of $S$ differ in exactly two places. Find the number of ways to order the letters in $H M M T T H E M E T E A M$ to get an almost palindrome.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Note that $T, E, A$ are used an odd number of times. Therefore, one must go in the middle spot and the other pair must match up. There are are $3 \cdot 2\left(\frac{6!}{2!}\right)=2160$ ways to fill in the first six spots with the letters $T, H, E, M, M$ and a pair of different letters. The factor of 3 accounts for which letter goes in the middle.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.