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Question #40634

Difficulty: 8/10 | 🟢 Free

The Question

Let $N$ be a random variable representing the number of heads obtained when 20 fair coins are tossed, given that at least one head appears. An integer $R$ is then chosen uniformly at random from the set $\{0, 1, 2, \dots, N\}$ . What is the probability that the binomial coefficient $\binom{N}{R}$ is the strictly maximum value in the set $\left\{ \binom{N}{k} : 0 \le k \le N \right\}$ ?

Options

A.
$\frac{3 \cdot 2^{20} - 23}{21(2^{20}-1)}$
B.
$\frac{2^{21} - 22}{21(2^{20}-1)}$
C.
$\frac{3 \cdot 2^{20} - 3}{21(2^{20}-1)}$
D.
$\frac{3 \cdot 2^{20} - 21}{21(2^{20}-1)}$

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

Greatest Binomial Coefficient|Combinatorial Probability|Complementary Counting

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle parity-based combinatorial properties and apply complementary counting rigorously.

The primary trap lies in assuming a unique maximum binomial coefficient for all $n$ , which leads to the incorrect numerator $2^{21}-22$ .

Furthermore, students who correctly partition the cases into even and odd $n$ often fail at the complementary counting step, forgetting to subtract the missing boundary terms $\binom{21}{1}$ and $\binom{21}{0}$ from the half-sums of the binomial expansion $2^{20}$ , leading to the other distractor options.

Quick Facts About This Question

Question Information

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