TITLE: At least how many students like all the activities?
QUESTION [5 upvotes]: I used Venn diagrams to solve the problem. It took me quite a while to find out the answer, and the answer is different from answer sheet. What is an easy way to solve the problem without getting lost in the middle of the process? According to the answer sheet, the correct answer is 32.

There are 100 students in a class. 75 of them like to play basketball. 80 like to play chess. 92 like to sing. 85 like to swim. At least how many students like all the 4 activities?

REPLY [2 votes]: Look at it this way. 25 don't play ball, 20, chess, 8 don't sing, 15 don't swim. This adds to 68, so at most, 68 people don't like one thing. The minimum that like all 4 is 32.