Let U be the set of all computers and let A be the set ofallcomputers used by 3 or more students. The problem asks to prove that |A| (the cardinality of A) is >= 5. Suppose this is not the case, i.e., |A| <= 4. Then |U \ A| = |U| - |A| = 12 - |A| >= 8. Also, each computer in U \ A is used by at most two students by the definition of A.

Again, k is the number ofLet be the number of computers used by or more students.[We must show that ] Because each computer is used by at most students, these computers are used by at most students(by the contrapositive form of the generalized pigeonhole principle). The remaining computers are each used by at most students.

Another question 2:

Why's that "The remaining computers are each used by at most students."?allcomputers with at least 3 users; the other 12 - k computers have at most 2 users by the definition of k.