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Principle of Inclusion and Exclusion

Count to 100. How many of those numbers are odd or multiples of 5? Since 50 are odd and 20 are multiples of 5, at first glance the answer is 70... See more

Generalized PIE


Determine the sum of all positive integers \(n \leq 100\) such that \(n\) is divisible by \(p^3\) for some integer \(p \geq 2\).

Each member of a \(60\)-player baseball team has at least one of the following \(4\) strengths: defense, base running, high batting average, and batting power. If the number of players good at base running is \(17,\) the number of players with high batting average is \(24,\) the number of players with batting power is \(13,\) the number of players who belong in two or more categories is \(12,\) the number of players who belong in three or more categories is \(8,\) and the number of players who belong in all four categories is \(2,\) how many players have defense as their strength?

Winston must choose 4 classes for his final semester of school. He must take at least 1 science class and at least 1 arts class. If his school offers 4 (distinct) science classes, 3 (distinct) arts classes and 3 other (distinct) classes, how many different choices for classes does he have?

Details and assumptions

He cannot take the same class twice.

Senior students at Brilliant High School are required to take at least one class in Geometry, Combinatorics, or Number Theory. They may take a class in more than one subject. If \(88\) student are taking Geometry, \(98\) students are taking Combinatorics, \(96\) students are taking Number Theory, \(11\) students are taking Geometry and Combinatorics, \(24\) students are taking Geometry and Number Theory, \(19\) students are taking Combinatorics and Number Theory, and \(6\) students are taking all the classes, how many students are in the senior class at Brilliant High?

Details and assumptions

When we say that \(88\) students are taking Geometry, these students may be taking Combinatorics and/or Number Theory as well.

Among \(80\) high school students, \(50\) students are enrolled in Trigonometry and \(37\) students are enrolled in World History. What is the minimum number of students who are enrolled in both Trigonometry and World History?


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