Question

Let $A$ be a $3\times 3$ matrix having entries from the set $\{–1,0,1\}$. The number of all such matrices $A$ having sum of all the entries equal to $5$, is .

Answer 1

Let matrix $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$
We need
$a+b+c+d+e+f+g+h+i=5$
$\begin{array}{cc}\text{Possible cases}& \text{Number of ways}\\ 5\to 1^{\prime }\text{s},4\to \text{zeroes}& \frac{9!}{5!4!}=126\\ 6\to 1^{\prime }\text{s},2\to \text{zeroes},1\to -1& \frac{9!}{6!2!}=252\\ 7\to 1^{\prime }\text{s},2\to -1^{\prime }\text{s}& \frac{9!}{7!2!}=36\end{array}$
Total ways $=126+252+36=414$