I'm trying to prove the following question from IMO 1972
Prove that for any positive integers $m,n$, $$\frac{(2m)!(2n)!}{m!n!(m+n)!}$$ is always an integer.
I tried thinking of a combinatorial argument, and came up with this: Imagine that there are two bags with $2m$ and $2n$ objects respectively. This expression is the number of ways of selecting $m$ objects from the first bag and $n$ objects from the second bag, such that the order in which you select these objects is not important. For instance, you can keep switching between bags, and also changing the order in which you select the objects.
However, I am not convinced that this is a correct argument. Where am I going wrong? Thanks