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Question #40661

Difficulty: 8/10 | 🟢 Free

The Question

Let $p_1, p_2, p_3, p_4, p_5$ be the first five prime numbers. Define a $5 \times 5$ matrix $M = [m_{ij}]$ such that $m_{ij} = \sum_{k=0}^5 (-1)^k \binom{p_i}{k} \binom{p_j}{5-k}$ for $1 \le i, j \le 5$ , where $\binom{n}{k} = 0$ if $n < k$ . Find the value of $\det(M)$ .

Options

A.
0
B.
217
C.
221
D.
5!

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Vandermonde's Identity|Skew-Symmetric Determinant|Transpose of Product|Elementary Row Operations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to identify hidden matrix properties within complex binomial summations, specifically targeting a variation of the Vandermonde identity structure.

The primary trap is the intimidating $5 \times 5$ determinant with prime-number-based binomial coefficients, which baits students into attempting brute-force expansion or complex row operations.

The 'null-trap' is triggered by recognizing that the index reversal in the summation proves the matrix is skew-symmetric.

Because the matrix is of odd order ( $n=5$ ), its determinant is inherently zero.

The numerical options (217, 221) serve as distractors to discourage guessing and penalize those who incorrectly expand partial submatrices.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Vandermonde's Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free