# Put some $1$ into an $n\times n$ grid (every check has at most one $1$) such that the number of $1$ in each row and each column is $k \ (<n)$

I want to ask if I can find a $1$ from each row such that for every two of $1$ they are not in the same column.

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Sure, put them all down the main diagonal. Or do you want the number of different ways to do it? – André Nicolas Sep 26 '12 at 0:32

There are $n!$ ways to put $n \ 1$'s into an $n \times n$ grid so that no two are in the same row or column. Start with the first row and put a $1$ in any cell-this gives $n$ choices. Now put a $1$ in the second row in some other column-$n-1$ choices. As you go down the rows, you have fewer and fewer choices until at the bottom you have only one choice, so you have $n(n-1)(n-2)\cdots1=n!$ ways to do it.
Let us count the ways. For the first row, choose any of the $n$ places. For everyone of these choices, the $1$ can be put in the second row in any one of $n-1$ places. For all the $(n)(n-1)$ ways of putting the $1$'s in the first two rows, there are $n-2$ places to put the $1$ in the third row. And so on. So the number of different configurations with the desired property is $n!$.