2021 AMC 10B Problem 4

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?

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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 .