Question

A university offers a Calculus class, a Sociology class, and a Spanish class. Youare given data below about a group of students who have all taken at least one ofthe three classes.(iv)(i) Group 1 contains 187 students. Of these, 61 students have taken Calculus,78 have taken Sociology, and 72 have taken Spanish. 15 have taken bothCalculus and Sociology, 20 have taken both Calculus and Spanish, and 13have taken both Sociology and Spanish. How many students in Calculushave taken all three classes?\

Answer 1

Let A, B, and C denote the sets of students that have taken calculus, sociology, and Spanish, respectively.

We're given that $A\cup B\cup C$ consists of 187 students.

We want to find the size of $A\cap B\cap C$, given that

• A has 61 students;

• B has 78;

• C has 72;

• $A\cap B$ has 15;

• $A\cap C$ has 20; and

• $B\cap C$ has 13.

Using the inclusion/exclusion principle, we have

$|A\cup B\cup C|=|A|+|B|+|C|-(|A\cap B|+|A\cap C|+|B\cap C|)+|A\cap B\cap C|$

where |X| denotes the size of the set X. Plug in all the known sizes:

$187=61+78+72-(15+20+13)+|A\cap B\cap C|$

$\implies|A\cap B\cap C|=24$

so 24 students have taken all three classes.