Question

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.)

$6$

$10$

$12$

$15$

$24$

Answer: D

Concepts:permutations counting pairs invariant

Difficulty rating: 1200

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.

2024 AMC 10A Problem 6

Answer: D

Difficulty rating: 1200

Solution:

Problem 6 in Other Years