AMC 10 2020b Problem 17

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? $\textbf{(A)}\ 11 \textbf{(B)}\ 12 \textbf{(C)}\ 13 \textbf{(D)}\ 14 \textbf{(E)}\ 15$