The number of matrices of order 3 × 3 , whose entries are either 0 or 1 and the sum of all the entries…

Let the matrix be $\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$

Since the elements can be either of $0 \text{or} 1$, so

$a+b+c+d+e+f+g+h+i=2/3/5/7$

For sum as $2$, the number of cases will be $\frac{9!}{2!7!}=36$

For sum as $3$, the number of cases will be $\frac{9!}{3!6!}=84$

For sum as $5$, the number of cases will be $\frac{9!}{5!4!}=126$

For sum as $7$, the number of cases will be $\frac{9!}{2!7!}=36$

Total required matrix will be $282$