Mathematics - Matrix Question with Solution | TestHub

Given,

A $5\times 5$ matrix whose each entry is either $0$ or $1$, is such that sum of entries of each column as well as each row is $1$,

Taking a possible case we get,

$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

Now arranging the above case we get,

$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]\left.\begin{array}{c}\to 5 \text{ways}\\ \to 4 \text{ways}\\ \to 3 \text{ways}\\ \to 2 \text{ways}\\ \to 1 \text{way}\end{array}\right\}$ because if we fix $1$ at first row and first column then in second row first column $1$ cannot come and vice-versa.

So total number of such matrices will be $5\times 4\times 3\times 2\times 1=120$