Problem:
There are 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 students taking yoga, taking bridge, and taking painting. There are students taking at least two classes. How many students are taking all three classes?
Answer Choices:
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Solution:
Let , and be the number of people taking exactly one, two, and three classes, respectively. The condition that each student in the program takes at least one class is equivalent to the equation . The condition that there are students taking at least two classes is equivalent to the equation . The sum counts once the students taking one class, twice the students taking two classes, and three times the students taking three classes. Then , which is equivalent to .
Let , and be the sets of students taking yoga, bridge, and painting, respectively. By the Inclusion-Exclusion Principle,
Furthermore, , because in tabulating the students taking at least two classes by considering the pairs of classes one by one, the students taking all three classes are counted three times rather than just once. Thus
so the number of students taking all three classes is .
The problems on this page are the property of the MAA's American Mathematics Competitions