Let $n=km$. Suppose we want to partition the set $\{1,2,\dots,n\}$ into $k$ blocks all of which have size $m$. My approach was the following (although I am thinking that an approach using exponential generating functions may be more appropriate):
First choose $m$ elements from $n=km$ in ${km\choose m}$ ways. Then choose $m$ more elements from the $km-m$ remaining elements and so on. This would lead to the number of partitions being $$\Pi_{j=0}^{k-1}{m(k-j)\choose m}=\Pi_{j=0}^{k-1}\left(m(k-j)\right)\left(m(k-j)-1\right)\dots\left(m(k-j)-m+1\right).$$
Is there a way to simplify this that I am not seeing? Or perhaps a different approach would be best? Thanks in advance.
