Problem:
Each of the 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 students who cannot sing, 65 students who cannot dance, and students who cannot act. How many students have two of these talents?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Because students cannot sing, can sing. Similarly, can dance, and can act. This gives a total of . However, the students with two talents have been counted twice in this sum. Because there are 100 students in all, students must have been counted twice.
OR
Consider the three sets referred to in the problem: those who cannot sing, those who cannot dance, and those who cannot act. Students with one talent are in two of those sets, whereas students with two talents are in only one. Thus the total counts all students twice except those with two talents. The number of students with two talents is therefore .
The problems on this page are the property of the MAA's American Mathematics Competitions