2021 AMC 12B Spring Problem 2

Below is the professionally curated solution for Problem 2 of the 2021 AMC 12B Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Spring solutions, or check the answer key.

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Concepts:basic counting parity

Difficulty rating: 1040

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?

$23$

$32$

$37$

$41$

$64$

Solution:

The $23$ all-blue pairs account for $46$ blue students, leaving $57 - 46 = 11$ blue students.

Each of those $11$ blue students must be paired with a yellow student, so there are $11$ mixed pairs, using $11$ yellow students.

The remaining $75 - 11 = 64$ yellow students form $64 \div 2 = 32$ all-yellow pairs.

Thus, the correct answer is B.

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