Number of ways to pick $d$ disjoint $k$-subsets out of $n$ elements

Question

How many ways are there to pick $d$ disjoint subsets of $k$ elements out of a set of $n$ elements? I got

$${n \choose dk}\prod_{i=0}^{d-1} {(d-i)k \choose k}$$

Is there a closed form? Does this require the dreaded... multinomial coefficient?

Answer 1

For an ordered set of $d$ $k$-subsets ($dk\le n$), it should be $$\frac{n!}{(k!)^d(n-dk)!}.$$ Thus, the number of unordered $d$ $k$-subsets is simply $$\frac{n!}{(k!)^d \,d!\,(n-dk)!}.$$

Answer 2

Wait, this is actually really easy. Also, the formula I gave in my question is wrong.

Namely. All you have to do is take all the permutations of the $dk$ elements, and then divide by $(k!)^d$, since each subset gets permuted, and then by $d!$, since the ordering of the subsets gets permuted. So it's $${n \choose dk}\frac{(dk)!}{(k!)^dd!}$$