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2017 AMC 10B Problem 13

Problem 13 of 25IntermediateAlgebraCounting & Probability

There are 2020 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 1010 students taking yoga, 1313 taking bridge, and 99 taking painting. There are 99 students taking at least two classes. How many students are taking all three classes?

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Solution

The number of classes taken total is 10+13+9=32.10+13+9=32. Let xx represent the number of people who take 1,1, let yy represent the number of people who take 22 classes, and let zz represent the number of people who take 33 classes. Then, we know x+2y+3z=32.x+2y+3z = 32. As such, the total number of people is 20,20, so x+y+z=20.x+y+z = 20. This makes y+2z=12.y+2z=12. The number of people who take at least two classes is 9,9, so y+z=9.y+z = 9. Therefore, z=3,z=3, making that the answer. Thus, the correct answer is C .

More practice

Concepts: inclusion-exclusion · system of equations

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