looking for a combinatorial interpretation given positive integers $n,m$ does the fraction
$$
\frac{(nm)!}{n!^mm!}
$$
count something? Namely does it correspond to the number of possibilities to do something?

Yes, it counts in how many ways that you can divide mn-elements set into m sets each with n elements.
For example we have got mn kids and they want to buy candies. But in the shop could be only n childs together (childrens are in n-elements groups). In how many ways childrens can be in the queue to the shop?
