Question

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?

$23$

$32$

$37$

$41$

$64$

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Written solution:

There are $2\cdot 23 = 46$ students with blue shirts that are in a pair with just blue shirts. This means there are $57-46=11$ students in blue shirts who are paired with someone wearing a yellow shirt, meaning exactly $11$ people wearing yellow shirts are paired with someone wearing a blue shirt.

This leaves just $64$ students wearing a yellow shirt who are paired with someone else wearing a yellow shirt. This yields $\dfrac{64}2 = 32$ pairs.

Thus, the answer is B .

2021 AMC 10B Spring Problem 4

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Problem 4 in Other Years