2025 AMC 12B Problem 13

Problem 13 of 25IntermediateCounting & Probability

A circle has been divided into

6

sectors of different sizes. Then

2

of the sectors are painted red,

2

painted green, and

2

painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below.

How many different colorings are possible?

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Solution

The two sectors of each color must be a non-adjacent pair, so a coloring is a way to split the

6

cyclic sectors into three non-adjacent pairs together with an assignment of the three colors. The non-adjacent pairs are the edges of the complement of the

6

-cycle, the triangular prism, which has

4

perfect matchings. Assigning the three colors in

3!

ways gives

4 \times 6 = 24

colorings. Thus, the correct answer is D.

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