5. a square board consists of $2 n \times 2 n$ squares. You want to mark $n$ of these squares so that no two marked squares are in the same or adjacent rows, and so that no two marked squares are in the same or adjacent columns. In how many ways is this possible?

## Solution:
We call a row or column selected if it contains a selected field. Of the $2 n$ rows, exactly $n$ are selected and no two of them are adjacent. If you number the rows from 1 to $2 n$, you can easily see that the marked rows have the numbers

$$
1,3, \ldots, 2 k+1,2 k+4, \ldots, 2 n-2,2 n
$$

where $0 \leq k \leq n$. The $n$ marked rows can therefore be selected in exactly $n+1$ ways. Similarly, the $n$ marked columns can be selected in $n+1$ ways. Finally, for given selected rows and columns, there are exactly $n$ ! possibilities to select the marked fields. The total answer is therefore $(n+1)^{2} n!$.