Let <em>A</em>, <em>B</em>, and <em>C</em> denote the sets of students that have taken calculus, sociology, and Spanish, respectively.
We're given that
consists of 187 students.
We want to find the size of
, given that
• <em>A</em> has 61 students;
• <em>B</em> has 78;
• <em>C</em> has 72;
•
has 15;
•
has 20; and
•
has 13.
Using the inclusion/exclusion principle, we have

where |<em>X</em>| denotes the size of the set <em>X</em>. Plug in all the known sizes:


so 24 students have taken all three classes.