Given the integers $[1, ck]$, they will be partitioned into $c$ subsets of size $k$. I want to count the number of unique versions of each subset (where order matters).
Clearly, there are ${ck \choose k}$ different assignments of integers to a given subset, and for a given assignment, there are $k!$ different orders for that assignment. However, I want to know the total number of unique versions of all subsets, I'm concerned that following the above reasoning for each subset independently, I may end up over counting.
It seems like there should be a simple combinatorial argument for what I want to count, so I suspect that this could be already answered in some other question, but I couldn't find it.