Problem Solving

A student of mine recently emailed me this question asking me how to solve it. It's a good GMAT question.

In a group of 68 students, each student is registered for at least one of three classes - History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?
A. 13 B. 10 C. 9 D.8 E. 7

This question can be sort of tricky. So we are given that there are 68 total students, but a total of 25+25+34=84 registrations for classes. We also know that there are 3 students registered for 3 classes, so these three students take up 3*3=9 registrations. This leaves us with with 84-9=75 registrations between 68-3=65 students. So we can set up an equation for this.

Let x be the number of students in 2 classes.

75=2x+65-x <<---Bc the students in 2 classes take up to registrations.
10=x

So we know there are 10 students in 2 classes, 65-10=55 students registered for 1 class and the 3 students enrolled in 3 classes for a total of 68 students.

As a check, we can multiply the registrations for each student and add them to make sure they total 84.

3 students enrolled in 3 classes = 9 registrations

10 students enrolled in 2 classes = 20 registrations

55 students enrolled in 1 class = 55 registrations
Total of 84 Registrations. Our answer of B, 10 students, checks.