Let A be a 3 $\times$ 3 matrix having entries from the set {$-$1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ___________.
Answer (integer)
414
Solution
<b>Case-I</b>: <br/><br/>$\begin{aligned} 1 & \rightarrow 7 \text { times } \\\\ \text { and }-1 & \rightarrow 2 \text { times } \end{aligned}$<br/><br/>
number of possible marrix $=\frac{9 !}{7 ! 2 !}=36$<br/><br/>
<b>Case-II</b>: <br/><br/>$1 \rightarrow 6$ times,<br/><br/>
$-1 \rightarrow 1$ times<br/><br/>
and $0 \rightarrow 2$ times<br/><br/>
number of possible marrix $=\frac{9 !}{6 ! 2 !}=252$<br/><br/>
<b>Case-III</b>: <br/><br/>$ 1 \rightarrow 5$ times,<br/><br/>
and $0 \rightarrow 4$ times<br/><br/>
number of possible marrix $=\frac{9 !}{5 ! 4 !}=126$<br/><br/>
Hence total number of all such matrix $A$ $=414$
About this question
Subject: Mathematics · Chapter: Permutations and Combinations · Topic: Fundamental Counting Principle
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