Let A be a 3 × 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 ______.
Correct Answer: 414
Solution and Explanation
Let matrix \(\begin{bmatrix} a & b & c \\[0.3em] d & e & f \\[0.3em] g & h & i \end{bmatrix}\)
Given, \(a + b + c + d + e + f + g + h + i = 5\)
| Possible cases | Number of ways |
|---|---|
| \(5 → 1’\)s, \(4 →\) zeroes | \(\frac {9!}{5!4!}=126\) |
| \(6 → 1’\)s, \(2 →\) zeroes, \(1 →–1\) | \(\frac {9!}{6!2!}=252\) |
| \(7 → 1’\)s, \(2 →–1'\)s | \(\frac {9!}{7!2!}=36\) |
Total number of ways \(= 126 + 252 + 36 = 414\)
So, the answer is \(414\).
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Concepts Used:
Permutations
A permutation is an arrangement of multiple objects in a particular order taken a few or all at a time. The formula for permutation is as follows:
\(^nP_r = \frac{n!}{(n-r)!}\)
nPr = permutation
n = total number of objects
r = number of objects selected
Types of Permutation
- Permutation of n different things where repeating is not allowed
- Permutation of n different things where repeating is allowed
- Permutation of similar kinds or duplicate objects