Difference between revisions of "2020 AMC 10B Problems/Problem 17"

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==Solution==
 
==Solution==
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Let us use casework on the number of diagonals.
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Case 1: <math>0</math> diagonals
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There are <math>2</math> ways: either <math>1</math> pairs with <math>2</math>, <math>3</math> pairs with <math>4</math>, and so on or <math>10</math> pairs with <math>1</math>, <math>2</math> pairs with <math>3</math>, etc.
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Case 2: <math>1</math> diagonal
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There are <math>5</math> possible diagonals to draw (everyone else pairs with the person next to them.
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Note that there cannot be 2 diagonals.
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Case 3: <math>3</math> diagonals
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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.
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Case 4: <math>5</math> diagonals
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There is <math>1</math> way to do this.
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Thus, in total there are <math>2+5+5+1=\boxed{13}</math> possible ways.
  
 
==Video Solution==
 
==Video Solution==

Revision as of 21:06, 7 February 2020

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

See Also

2020 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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