JEE Main 2022 (Online) 25th June Morning ShiftQuestion 1
Let $A$ be a $3\times3$ 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 $\_\_\_\_\_\_$ .
Permutation and Combination
Answer$414$
Solution
Let $p$ ones, $q$ minus-ones, $p-q=5$, $p+q\le9$: $(5,0),(6,1),(7,2)$. Counts: $\binom95=126$; $\binom96\binom31\cdot$ ... $=84\cdot3=252$; $\binom97\binom22=36$. Total $126+252+36=414$.