Euler's Officers

This has proved to be a Tough Nut. Reading the article on Latin Squares published in September 2002 should help you to solve this.

Taking $s=1$, $2$, $3$ or $4$ you can construct $4$ different Latin squares $S^{i,j}$ of order $5$ where $S^{i,j}=si+j \pmod 5$.

Now suppose the numbers are used to denote the five ranks and consider how many different arrangements there will be if no two officers of the same rank or of the same regiment appear in the same row or in the same column.