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Question #40562

Difficulty: 8/10 | 🟢 Free

The Question

Let $a = \sum_{r=0}^{1} \binom{2}{r} \binom{2}{1-r}$ and $b = \sum_{r=0}^{2} (-1)^r \binom{2}{r} 2^{2-r}$ . A multiset $S$ consists of exactly $a$ identical objects of type $A$ and $b$ identical objects of type $B$ . Let $k$ be the total number of distinct linear arrangements of all the objects in $S$ .

An experiment consists of $n$ independent trials, where $n$ is the constant term in the binomial expansion of $\left(x + \frac{1}{x}\right)^4$ . In each trial, the probability of success is $p = \frac{b}{a+b}$ .

If $P$ is the probability of obtaining at least $k$ successes in $n$ trials, then $P$ equals:

Options

A.
$\frac{1}{625}$
B.
$\frac{24}{15625}$
C.
$\frac{1}{15625}$
D.
$\frac{624}{625}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Permutations of Identical Objects|Properties of Binomial Coefficients|Binomial Distribution Tail-Ends

Why This Question Is Tricky

Common Mistakes

This question constructs a 'Lethal Collision' by chaining combinatorial identities into a Binomial Distribution tail-end calculation.

The student must first correctly evaluate a binomial convolution (Vandermonde's identity) and a binomial expansion to find the multiset frequencies.

The primary trap lies in the final probability stage: students frequently misinterpret 'at least $k$ successes' and calculate only the exact probability $P(X=k)$ , leading directly to the trap option $\frac{24}{15625}$ .

Additionally, any error in the preceding combinatorial steps (such as miscalculating the constant term $n$ or the permutation count $k$ ) will catastrophically propagate, ensuring only those with complete mastery of both combinatorics and probability reach the correct answer.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Permutations of Identical Objects
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free