Problem
COM-B2-M01-P002 A Club with Many Members
#2
★★☆☆☆ Level 2 of 5
A school has \(120\) students and \(15\) clubs. Each student attends exactly \(4\) clubs. Prove that some club has at least \(32\) students.
Count pairs \((student, club)\).
The number of pairs \((student, club)\) where the student attends the club is \(120\cdot4=480\). If every club had at most \(31\) members, there would be at most \(15\cdot31=465\) such pairs, a contradiction. Hence some club has at least \(32\) members.
This is an average argument with integer rounding.