Number of ways to place $k$ non-attacking rooks on an $m\times n$ chessboard I need to calculate the number of ways to place $k$ non-attacking rooks on an $m \times n$ table where $k \leq n$ and $k \leq m$.  ("Non-attacking" means that no two rooks may share a row or column.)  My attempt: 
Calculate the number of ways to place $k$ rooks on a $k \times k$ board ($k!$), then multiply by the number of ways to select a $k \times k$ board from an $m \times n$ board. (This is the part I can't calculate, if it is correct at all.) 
My question: 
Is my approach good and if so, how to calculate the second part?
It is a reasonable approach. The columns can be chosen in $\binom{m}{k}$ ways  and for each way of selecting columns the rows can be chosen in $\binom{n}{k}$ ways.
I think you can choose $k$ squares out of $nm$ in $\pmatrix{ nm\\k}$ different ways, and for each one of these choices there are $k!$ different ways to set the rooks, so the result is $k!\pmatrix{ nm\\k}$.