Euler's Officers

How many different solutions can you find to this problem?

Arrange 25 officers, each having one of five different ranks $a$, $b$, $c$, $d$ and $e$, and belonging to one of five different regiments $p$, $q$, $r$, $s$ and $t$, in a square formation 5 by 5, so that each row and each file contains just one officer of each rank and just one from each regiment.