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Question #41849

Difficulty: 8/10 | 🟢 Free

The Question

A $3 \times 3$ matrix $M = [m_{ij}]$ is constructed by choosing its entries independently and uniformly at random from the set of all distinct coefficients of $x^k$ ( $0 \le k \le 5$ ) in the expansion of $(1 - x)^5$ . Let $C_{ij}$ and $M_{ij}$ denote the cofactor and minor of the element $m_{ij}$ , respectively. What is the probability that $C_{12} > M_{12}$ ?

Options

A.
$\frac{49}{108}$
B.
$\frac{11}{27}$
C.
$\frac{25}{54}$
D.
0

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Multiplication Principle Scope|Combinatorial Counting (P&C Integration)|Coefficient of x^r in Expansion|Minors and Cofactors Sign Convention

Why This Question Is Tricky

Common Mistakes

This question tests the integration of combinatorial counting, probability, and matrix properties.

The primary logical trap lies in the sign convention of cofactors: students often incorrectly assume $C_{ij} = M_{ij}$ or fail to apply the $(-1)^{i+j}$ factor, which would lead them to conclude the probability is 0.

Furthermore, it tests the ability to efficiently calculate the probability of $Z_1 < Z_2$ for symmetric discrete random variables by leveraging the identity $P(Z_1 < Z_2) = (1 - P(Z_1 = Z_2))/2$ , avoiding exhaustive and error-prone manual counting of favorable cases.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Multiplication Principle Scope
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free