2020 AMC 12B Problems/Problem 15

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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$$ (Error compiling LaTeX. Unknown error_msg)(B) 12$$ (Error compiling LaTeX. Unknown error_msg)(C) 13$$ (Error compiling LaTeX. Unknown error_msg)(D) 14$$ (Error compiling LaTeX. Unknown error_msg)(E) 15$