A midterm exam consists of $5$ problems. Students who solve two of those problems correctly get a passing grade. There are $32$ students in the class, and only $8$ students passed. Prove that one of the problems was solved correctly by at most $12$ students.
So $16$ questions were solved correctly, so at least one problem was solved at least $3$ times by the pigeon hole principle. Where do the $12$ students come from? I mean can't the $24$ students solve $0$ questions or do I assume that at least each student solve one question? Thanks.