Sets of students, unions, complements, intersections

Question

There are 93 students in the class; 42 like Math, while 41 like English. If 30 students don't like either subject, how many students like both?

A.10

B.20

C.41

D. The answer cannot be determined from the data given.

Answer 1

Hint:

The principle of inclusion-exclusion tells us that

$$|A\cup B|=|A|+|B|-|A\cap B|$$

Let $M$ represent the set of students in the class who like math and $E$ who like english.

The problem tells you $|M|$ and $|E|$ directly and gives you enough information to find $|M\cup E|$ (the amount of people who like at least one of math or english)

People who like at least one of math or english are those people who aren't a part of the people who don't like either.

The problem is now to use this information to find $|M\cap E|$, the amount of students who like both.

It may help to draw yourself a Venn Diagram to help visualize the information.