JEE MAIN - Mathematics (2022 - 25th June Morning Shift - No. 20)
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
414
Explanation
Case-I:
$\begin{aligned} 1 & \rightarrow 7 \text { times } \\\\ \text { and }-1 & \rightarrow 2 \text { times } \end{aligned}$
number of possible marrix $=\frac{9 !}{7 ! 2 !}=36$
Case-II:
$1 \rightarrow 6$ times,
$-1 \rightarrow 1$ times
and $0 \rightarrow 2$ times
number of possible marrix $=\frac{9 !}{6 ! 2 !}=252$
Case-III:
$ 1 \rightarrow 5$ times,
and $0 \rightarrow 4$ times
number of possible marrix $=\frac{9 !}{5 ! 4 !}=126$
Hence total number of all such matrix $A$ $=414$
$\begin{aligned} 1 & \rightarrow 7 \text { times } \\\\ \text { and }-1 & \rightarrow 2 \text { times } \end{aligned}$
number of possible marrix $=\frac{9 !}{7 ! 2 !}=36$
Case-II:
$1 \rightarrow 6$ times,
$-1 \rightarrow 1$ times
and $0 \rightarrow 2$ times
number of possible marrix $=\frac{9 !}{6 ! 2 !}=252$
Case-III:
$ 1 \rightarrow 5$ times,
and $0 \rightarrow 4$ times
number of possible marrix $=\frac{9 !}{5 ! 4 !}=126$
Hence total number of all such matrix $A$ $=414$
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