Within a grade, say that $20$ people are taking math, $23$ are taking biology, $29$ are taking economics, and 31 are taking communications. If there are $32$ people in the grade, prove that there are at least $2$ people who are taking all $4$ classes.
I would really appreciate any help on this, I've been adding all the individual class counts and then dividing by the number of people in the grade, but I don't think that's the proper way to prove it!
If we add all the students in each class we have $103$ enrollments. Now if all $32$ took exactly three classes each, the maximum that can be obtained without someone taking four classes, we would only have $32 \times 3 = 96$ enrollments. So if we minimize the number of students required to take four classes we still need at least $7$ students to take four classes. Since $7 > 2$ the result follows.