What is the number of all matrices $n*n$ with entries in $ \{0,1\}$ that in every row and every column has at least one 1?
I think maybe it is easier to count those matrices that has some null-row or null-columns. Any ideas?
Thanks in advance.
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If $A_n$ is that number, then: Choose the position of the $1$ in the first column: $n$ possibilities. Delete the column and the row of this $1$ and group together the $n-1\times n-1$ matrix that remains. The number of ways to solve the problem for that smaller matrix is $A_{n-1}$. Observe that any solution for the smaller problem of size $n-1$ yields a solution for the original of size $n$. – orole Jan 29 '18 at 19:40
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Maybe to define sets: $A_1= $ all matrices where first row is null-row,$ A_2=$ all matrices where second row is null-row,... $A_n=$ matrices where n-th row is null row. Then our set of matrices is set of all those matrices ( there are $2^{n^2}$ of them) minus set $A_1 \cup A_2 \cup...A_n$ ? – zariski Jan 29 '18 at 19:44