2016 AMC 12A Problem 11

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.