2024 AMC 10A Problem 6

What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA? (For example,

3

swaps are required to change ABC to CBA; one such sequence of swaps is ABC

\to

BAC

\to

BCA

\to

CBA.)

Show solution

Solution

Reversing all six letters flips the relative order of every pair, so all

\binom{6}{2} = 15

pairs end up inverted. Each adjacent swap fixes exactly one inversion. So we need at least

15

swaps, and bubbling each letter into place hits

15

exactly. Therefore, the answer is D.