2020 AMC 12B Problems/Problem 15

Problem

There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him,as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know each other?

$(A)$ $11$ $(B)$ $12$ $(C)$ $13$ $(D)$ $14$ $(E)$ $15$

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2020 AMC 12B Problems/Problem 15

Problem