Skip to main content

2016 AMC 12A Problem 11

Problem 11 of 25IntermediateCombinatorics

Each of the 100100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 4242 students who cannot sing, 6565 students who cannot dance, and 2929 students who cannot act. How many students have two of these talents?

Answer choices

Show solution

Solution

The numbers who can sing, dance, and act are 100−42=58,100-42=58, 100−65=35,100-65=35, and 100−29=71,100-29=71, respectively, for a total of 58+35+71=164.58+35+71=164. Since no student has all three talents, each student has one or two talents, so single-talent students are counted once and two-talent students are counted twice. The number counted twice is 164−100=64.164-100=64. Thus, the correct answer is E.
AoPS wiki

Tagged: complementary counting · inclusion-exclusion · double counting

More practice