In a school, there are $m$ teachers and $n$ students. Suppose each teacher has exactly $k$ students, and each pair of distinct students has exactly $\ell$ teachers. Determine a relation between $m, n, k, \ell$.
Let $X$ be the number of triplets $\left(e_{1}, e_{2}, p\right)$ where $e_{1}$ and $e_{2}$ are two different students who have $p$ as their teacher. We have $|X|=m k(k-1)=n(n-1) \ell$.