Skip to main content

2021 AMC 12B Problem 2

Problem 2 of 25EasierNumber TheoryCounting & Probability

At a math contest, 5757 students are wearing blue shirts, and another 7575 students are wearing yellow shirts. The 132132 students are assigned into 6666 pairs. In exactly 2323 of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?

Answer choices

Show solution

Solution

The 2323 all-blue pairs account for 4646 blue students, leaving 5746=1157 - 46 = 11 blue students. Each of those 1111 blue students must be paired with a yellow student, so there are 1111 mixed pairs, using 1111 yellow students. The remaining 7511=6475 - 11 = 64 yellow students form 64÷2=3264 \div 2 = 32 all-yellow pairs. Thus, the correct answer is B.

More practice

Concepts: basic counting · parity

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.