I suppose that this question has already been asked, but I couldn't find it. Suppose we have a set $A$ with $nk$ elements. How many partitions of this set into sets of k elements are there?. For $n=k=2$, there are 3:
$\{\{\{1,2\}, \{3,4\}\}, \{\{1,3\}, \{2,4\}\}, \{\{1,4\},\{2,3\}\}$
NOTE: The case $n=2$ is solved here: Counting the number of partitions having blocks of cardinality 2 and non-distinct elements