If m and n are positive integers more than or equal to 2,$m>n,$ then $\left(mn\right)$ ! is divisible by

We know that $\frac{\left(mn\right)!}{(m!{)}^n}$  is the number of ways of 

distributing $mn$ distinct object inn persons equally.

Hence, $\frac{\left(mn\right)!}{(m!{)}^n}$ is a positive integer.

Consequently, $\left(mn\right)$ ! is divisible by $(m!{)}^n$

Similarly, $\left(mn\right)$ ! is divisible by $(n!{)}^m$

Now,

$\begin{array}{l}n<m\Rightarrow m+n<2m\le mn\text{ and }m-n<m<mn\\ \Rightarrow (m+n)!\mid \left(mn\right)!\text{ and }(m-n)!\mid \left(mn\right)!\end{array}$