In how many ways can $mn$ things be distributed equally among $n$ groups?
My logic - I simply thought of the answer as $mn/n$ (i.e. in the same way we divide 15 chocolates among $n$ students). But obviously my answer is wrong. please guide me.
Answer: $\frac{mn!}{m!^n}$.
$\frac {mn}n=m$ is correct for the number of objects each person gets, but not for the ways to choose which objects each person gets. One way is to line the objects up, which you can do in $(mn)!$ ways, then give the first person the first $m$, the second person the next $m$ and so on. The first person's objects can come in any of $m!$ orders, so that many of the initial orderings are equivalent. The same goes for each other person, so we need to divide by $(m!)^n$ giving the answer you quote:$$\frac {(mn)!}{(m!)^n}$$