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Question #41067

Difficulty: 8/10 | 🟢 Free

The Question

Let $k$ be the maximum power of $x$ in the numerically greatest term(s) in the expansion of $(1 + x)^{13}$ when $x = \frac{3}{4}$ . Alice tosses a fair coin $k$ times and Bob tosses a fair coin $k+2$ times. Let $P$ be the probability that Alice and Bob obtain the same number of heads. Find the value of $P$ .

Options

A.
$\frac{99}{512}$
B.
$\frac{3003}{16384}$
C.
$\frac{715}{4096}$
D.
$\frac{429}{2048}$

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

Combinatorial Counting (P&C Integration)|Vandermonde's Identity|Binomial Coefficient Summation Properties|Numerically Greatest Term

Why This Question Is Tricky

Common Mistakes

The problem tests the student's ability to correctly identify the Numerically Greatest Term (NGT) when the ratio $T_{r+1}/T_r = 1$ yields an integer, creating two equal greatest terms.

The trap lies in failing to account for this boundary condition, leading students to incorrectly select $k=5$ (the power of the first NGT) or $k=7$ (the term number instead of the power).

Furthermore, it integrates this combinatorial parameter into a probability sequence, requiring the application of Vandermonde's Identity ( $\sum \binom{A}{k-m}\binom{B}{m} = \binom{A+B}{k}$ ) to resolve the summation of binomial products.

Failure at the NGT stage cascades into an incorrect Vandermonde summation, perfectly matching the incorrect options provided.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Combinatorial Counting (P&C Integration)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free