Number of matrices with each row and column having exactly one 1.

Consider a square matrix of order $$n = 5$$ such that $$a_{ij} = 0 ~ \forall ~ i+j = n+1; a_{ij} \in \{0,1\}$$. In each row as well as in each column there is only one non zero element. Then number of such matrices is?

First we note that right diagonal has only $$0$$s.

Then I tried it this way: We choose a place for one among each row and mark the other places in that column and same row as forbidden (i.e. no more one's).

So for first column we have 4 choices then 3 choices then 2 then 1 and then 2s.

Thus, Number of ways = $$4\times 3 \times 2\times 1\times 2 = 48$$

But its erroneous because the number of choices change if we place 1 above 0 in each attempt.

What's the correct way to solve this question?

Flip the array uoside down. Then $$a_{ii}=0$$, so the positions of the nonzero $$a_{ij}$$ form a derangement of the numbers from $$1$$ to $$5$$.
• Suppose the nonzero cells are $a_{1a},a_{2b},a_{3c},a_{4d}$ and $a_{5e}$. Then $a,b,c,d,e$ are five different numbers, and $1\neq a,2\neq b,3\neq c,4\neq d,5\neq e$. – Empy2 Apr 3 at 8:47
• The only one in row 1 is $a_{1a}$. The only one in column $a$ is $a_{1a}$. – Empy2 Apr 3 at 9:49