Problem:
Find the number of ways in which the letters in "HMMTHMMT" can be rearranged so that each letter is adjacent to another copy of the same letter. For example, "MMMMTTHH" satisfies this property, but "HHTMMMTM" does not.
HMMT November, 2021
Solution
Solution:
The final string must consist of "blocks" of at least two consecutive repeated letters. For example, MMMMTTHH has a block of 4 M's, a block of 2 T's, and a block of 2 H's. Both H's must be in a block, both T's must be in a block, and all M's are either in the same block or in two blocks of 2. Therefore all blocks have an even length, meaning that all we need to do is to count the number of rearrangements of the indivisible blocks "HH", "MM", "MM", and "TT". The number of these is .
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