AUSTRIAN-POLISH 1997 Problem 2

Each square of an $n \times m$ board is assigned a pair of coordinates $(x,y)$ with $1 \le x \le m$ and $1 \le y \le n$ . Let $p$ and $q$ be positive integers. A pawn can be moved from the square $(x,y)$ to $(x',y')$ if and only if $|x - x'| = p$ and $|y- y'| = q$ . There is a pawn on each square. We want to move each pawn at the same time so that no two pawns are moved onto the same square. In how many ways can this be done?