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Question #40653

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_m(x) = \sum_{k=1}^n k^m \binom{n}{k} x^k$ for $m \in \mathbb{N}$ and $x > 0$ . A function $f(x)$ is defined as $f(x) = \frac{S_3(x)}{S_1(x)} - \left(\frac{S_2(x)}{S_1(x)}\right)^2$ . If the maximum value of $f(x)$ for $x \in (0, \infty)$ is $20$ , then the value of $n$ is:

Options

A.
79
B.
80
C.
81
D.
82

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

Properties of Binomial Coefficients|Variance of Discrete Variable|Bernoulli Trials Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map complex combinatorial sums to the moments of a probability distribution, specifically the variance of a size-biased binomial distribution.

The primary trap is the 'Bernoulli Trial' dependency: students often attempt to evaluate $S_1, S_2, S_3$ through brute-force differentiation of $(1+x)^n$ or algebraic factorial expansion.

This leads to massive time consumption and a high probability of calculation errors in the second and third moments.

By recognizing $f(x)$ as $\text{Var}(X) = E[X^2] - (E[X])^2$ for a shifted binomial random variable $X-1 \sim B(n-1, \frac{x}{x+1})$ , the variance $(n-1)pq$ can be written directly.

The distractor 80 catches students who forget the index shift and incorrectly use $n$ instead of $n-1$ in the variance formula.

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