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

Case-I: $ 1 \rightarrow 7$ times
and $-1 \rightarrow 2$ times
number of possible matrix $=\frac{9 !}{7 ! 2 !}=36$
Case-II: $ 1 \rightarrow 6$ times,
$-1 \rightarrow 1$ times
and $0>2$ times
number of possible matrix $=\frac{9 !}{6 ! 2 !}=252$
Case-III: $ 1 \rightarrow 5$ times,
and $0 \rightarrow 4$ times
number of possible matrix $=\frac{9 !}{5 ! 4 !}=126$
Hence total number of all such matrix A $=414$