Question 14
In a school, there are totally $n$ students, with $n \geq 2$. The students take part in $m$ clubs and in each club, there are at least $2$ members (a student may take part in more than $1$ club). Eventually, the Principal notices that: If $2$ clubs share at least $2$ common members then they have different numbers of members. Prove that
$$
m \leq (n-1)^2.
$$
$$
m \leq (n-1)^2.
$$