Question

Each of the $100$ 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 $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents?

$16$

$25$

$36$

$49$

$64$

Answer: E

Concepts:complementary counting inclusion-exclusion double counting

Difficulty rating: 1470

Small Hint:

Count how many students can do each activity

Big Hint:

Adding the three counts tallies two-talent students twice; subtract $100$

Solution:

The numbers who can sing, dance, and act are $100-42=58,$ $100-65=35,$ and $100-29=71,$ respectively, for a total of $58+35+71=164.$

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 $164-100=64.$

Thus, the correct answer is E.

2016 AMC 12A Problem 11

Answer: E

Difficulty rating: 1470

Small Hint:

Big Hint:

Solution:

Problem 11 in Other Years