XXII OM - II - Problem 1

In how many ways can $ k $ fields of a chessboard $ n \times n $ ($ k \leq n $) be chosen so that no two of the selected fields lie in the same row or column?
First, one can choose $ k $ rows in which the selected fields will lie. This can be done in $ \displaystyle \binom{n}{k} $ ways. Then, in each of these rows, one must sequentially choose one of $ n $ fields, one of $ n - 1 $ fields lying in the remaining columns, etc. - one of $ n - k + 1 $ fields. Therefore, the total number of ways is $ \displaystyle \binom{n}{k} n(n - 1) \ldots (n - k + 1) $.