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, 2015

Combinatorics Difficulty 5.0 AIME, harder Prove it United States

Problem:
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 HMMTTHEMETEAM to get an almost palindrome.

Solution

Solution:
Answer: $2160$
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 $3 \cdot 2 \left(\frac{6!}{2!}\right) = 2160$ 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: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.