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Question #40652

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} \binom{x}{1} & \binom{x}{2} & \binom{x}{3} & 1 \\ \binom{2}{1} & \binom{2}{2} & \binom{2}{3} & 1 \\ \binom{4}{1} & \binom{4}{2} & \binom{4}{3} & 1 \\ \binom{6}{1} & \binom{6}{2} & \binom{6}{3} & 1 \end{vmatrix}$ . Find the value of $\sum_{x=7}^{10} f(x)$ .

Options

A.
480
B.
-480
C.
360
D.
-360

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Properties of Binomial Coefficients|Factor Theorem in Determinants|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Factor Theorem in determinants alongside the properties of Vandermonde matrices.

The primary trap lies in the column exchange parity: students often correctly identify the roots and the Vandermonde structure but fail to account for the $(-1)^3$ sign change when cyclically permuting the columns to reach the standard form.

This leads to an inverted sign in the final polynomial.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Properties of Binomial Coefficients
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free