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Question #40720

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the rank of the word 'RANK' in the dictionary order among all permutations of its letters. Evaluate the sum: $S = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \sum_{k=0}^{n} \binom{R}{k} \binom{R}{n-k}$

Options

A.
$\sum_{j=1}^{40} \frac{1}{j}$
B.
$\sum_{j=1}^{20} \frac{1}{j}$
C.
$\binom{40}{20} \ln 2$
D.
$\sum_{j=1}^{39} \frac{1}{j}$

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

Logarithmic Series|Vandermonde's Identity|Rank of a Word

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of discrete combinatorics and continuous calculus.

The primary trap lies in the infinite upper limit of the outer sum.

Students often incorrectly assume the series evaluates to a logarithmic constant (like $\ln 2$ ) multiplied by a combinatorial factor, failing to recognize that the Vandermonde convolution acts as a combinatorial filter.

This filter naturally truncates the infinite series at $n=2R$ .

Furthermore, evaluating the resulting finite sum requires mapping the discrete binomial coefficients to the continuous integral of a geometric progression, ultimately yielding a Harmonic number rather than a natural logarithm.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Logarithmic Series
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free