2021 AMC 10B Problems/Problem 4

Problem

At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The 132 students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?

$\textbf{(A)} ~23 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~37 \qquad\textbf{(D)} ~41 \qquad\textbf{(E)} ~64$

Solution

There are $46$ students paired with a blue partner. The other $11$ students wearing blue shirts must each be paired with a partner wearing a shirt of the opposite color. There are $64$ students remaining. Therefore the requested number of pairs is $\tfrac{64}{2}=\boxed{\textbf{(B)} ~32}$ ~Punxsutawney Phil

Video Solution by OmegaLearn (System of Equations)

https://youtu.be/hyYg62tT0sY

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/gLahuINjRzU?t=626

~IceMatrix

Video Solution by Interstigation

https://youtu.be/DvpN56Ob6Zw?t=286

~Interstigation

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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All AMC 10 Problems and Solutions

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