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Question #40640

Difficulty: 8/10 | 🟢 Free

The Question

Let $n$ be the number of ways to distribute 20 identical items among 3 distinct boxes such that exactly one box is empty. Consider the binomial expansion of $(5 - 2x)^n$ in increasing powers of $x$ . If $x = 2$ , let $k$ be the index of the algebraically greatest term in the expansion. Find the value of $k$ .

Options

A.
25
B.
26
C.
27
D.
29

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

Numerically Greatest Term|Distributing Identical Items (Stars and Bars)|Constant Function Logic|Sign Flipping Rule

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between the Numerically Greatest Term (NGT) and the algebraically greatest term in a binomial expansion with a negative parameter.

Students typically calculate the NGT index using the standard formula $m = (n+1)|x| / (|a|+|x|)$ , which yields $m \approx 25.77$ , pointing to $T_{26}$ .

The primary trap is failing to apply the 'Sign Flipping Rule': because the term is negative, $T_{26}$ is actually the algebraically least term.

A secondary trap occurs when students realize the term must be positive (odd index) and mistakenly choose $T_{25}$ because its index is numerically closer to $25.77$ than $27$ .

However, due to the skewed binomial distribution, $T_{27}$ is strictly greater than $T_{25}$ , forcing students to analytically compare the adjacent positive terms rather than relying on proximity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Numerically Greatest Term
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free