[Math] How many ways of splitting objects equally into groups

Question

How many ways in which $m\cdot n$ distinct objects can be divided equally into $n$ groups?

The answer is $$\frac{(mn)!}{(m!)^n n!}$$

Can someone please supply the intuition behind this answer?

Thanks in advance.

Answer 1

Imagine groups are written down in a row. This is same as permuting the original $n\cdot m$ objects and assigning the first $m$ object to the first group, the second $m$ objects to second group etc.

Now each such group has $m!$ ways it can be permuted, so there are $(m!)^n$ permutations that give the same groups.

Hence the answer $$ \frac{(m\cdot n)!}{(m!)^n} $$

You need to fill in the gaps!