2020 AMC 10B Problems/Problem 17

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?

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

Solution

Let us use casework on the number of diagonals.

Case 1: $0$ diagonals There are $2$ ways: either $1$ pairs with $2$ , $3$ pairs with $4$ , and so on or $10$ pairs with $1$ , $2$ pairs with $3$ , etc.

Case 2: $1$ diagonal There are $5$ possible diagonals to draw (everyone else pairs with the person next to them.

Note that there cannot be 2 diagonals.

Case 3: $3$ diagonals

Note that there cannot be a case with 4 diagonals because then there would have to be 5 diagonals for the two remaining people, thus a contradiction.

Case 4: $5$ diagonals There is $1$ way to do this.

Thus, in total there are $2+5+5+1=\boxed{13}$ possible ways.

Video Solution

https://youtu.be/3BvJeZU3T-M

~IceMatrix

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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All AMC 10 Problems and Solutions

2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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All AMC 12 Problems and Solutions

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2020 AMC 10B Problems/Problem 17

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Solution

Video Solution

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