Question

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?

Answer 1

there are 10 people standing equally spaced around a circle. each person knows exactly 3 of the other 9 people then there are 10 possible ways

there are 10 people standing equally spaced around a circle. each person knows exactly 3 of the other 9 people:

There is only $1$ way to do this.

Thus, in total there are $2+5+5+1={(C) }13}$ possible ways.

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