Track / Stage 4 / 332 of 340 #592 of 1964

Problem 592

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Find the answer

In a group of 40 students, 20 play football, 19 play volleyball, and 15 play exactly one of these two sports. How many students do not practice either football or volleyball?
(a) 7
(b) 5
(c) 13
(d) 9
(e) 10

Your answer

Multiple choice: answer with the letter of the option you want.

Official solution

The correct option is (c).

Let $x$ be the number of students who practice both sports. Therefore, the number of students who practice only football is $20-x$ and the number who practice only volleyball is $19-x$ . Thus, the 15 students who practice exactly one sport are divided into $15=(20-x)+(19-x)$ . It follows that $x=12$ and it results that $20+(19-x)=27$ students practice at least one of the two sports. Therefore, $13=47-27$ students do not practice either football or volleyball.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.