# Find the number of $n × n$ binary matrices that have precisely one 1 in every row and every column. [closed]

We want to find the number of $$n × n$$ binary matrices that have precisely one 1 in every row and every column. I believe the number of such matrices is $$n!$$. This is my reasoning: the number of $$n × n$$ binary matrices that have precisely one 1 in every row and every column corresponds to all possible permutations of the identity matrix (i.e. every permutation matrix of size $$n$$), of which there are $$n!$$. How would one go about proving this?

## closed as off-topic by Namaste, Leucippus, Xander Henderson, Jendrik Stelzner, user91500Aug 23 '18 at 9:13

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• Do you ever hear about permutation? – Cloud JR Aug 22 '18 at 13:32
• Yes. I'm assuming the number of such matrices would correspond to all possible permutations of the identity matrix, correct? – Frank Aiello Aug 22 '18 at 13:33
• You are correct – Cloud JR Aug 22 '18 at 13:38

## 1 Answer

The first row can be filled in $n$ ways ($n$ columns is possible to put $1$ and $0$ in other column ). After filling first row there is $(n-1)$ ways (One column is already used) as so on.

BY FUNDAMENTAL PRINCIPLE OF COUNTING, there are $n!$ ways.