MathDB
n students in m clubs

Source: KJMO 2014 p8

May 2, 2019
combinatoricsSets

Problem Statement

Let there be nn students and mm clubs. The students joined the clubs so that the following is true: - For all students xx, you can choose some clubs such that xx is the only student who joined all of the chosen clubs. Let the number of clubs each student joined be a1,a2,...,ama_1,a_2,...,a_m. Prove that a1!(m−a1)!+a2!(m−a2)!+...+an!(m−an)!≤m!a_1!(m - a_1)! + a_2!(m - a_2)! + ... + a_n!(m -a_n)! \le m!
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