partitioning of a set with kn members into k subsets such that each subset has n members

Question

we know that $S(n,k)$ is the number of ways we can partition a set with $n$ members to $k$ subsets ( each subset has at least one member).
imagine we have a set with $k*n$ members. we want to partition this set to $k$ subsets such that each of them has $n$ members. how many ways can we do this ?

Answer 1

This is a multinomial coefficient "in disguise" and has little to do with the Stirling numbers.

Imagine that the set elements are arranged in some sequence, and we take the subsets of size $n$ as they appear from left to right. Permutations of the $k$ blocks of size $n$ do not change the partition, nor do permutations within any of the $k$ blocks of size $n$.

It follows that the count is:

$$ \frac{(kn)!}{k! (n!)^k} $$