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Question #40597

Difficulty: 8/10 | 🟢 Free

The Question

Four distinct energy states can each hold a maximum of $5$ identical quantum particles. $14$ such identical particles are distributed among these $4$ states. Let $S$ be the sample space of all valid distributions. A distribution is selected at random from $S$ . Let $E$ be the event that the selected distribution has \textit{exactly one} energy state containing exactly $5$ particles.

If this selection process is repeated independently with replacement, what is the probability that the $3^{\text{rd}}$ occurrence of event $E$ happens exactly on the $6^{\text{th}}$ selection?

Options

A.
$\frac{5}{32}$
B.
$\frac{5}{16}$
C.
$\frac{15}{64}$
D.
$\frac{3}{32}$

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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

Selection from Identical Items|Inclusion-Exclusion Principle|Binomial Distribution Tail-Ends|Sample Space Partitioning

Why This Question Is Tricky

Common Mistakes

The question tests the intersection of constrained combinatorial distributions (Stars and Bars with Inclusion-Exclusion) and Negative Binomial probability.

The primary trap lies in 'Sample Space Partitioning' and boundary conditions: students often miscalculate the restricted sample space $|S|$ by failing to properly apply inclusion-exclusion, or they miscalculate $|E|$ by failing to apply the strict $x_i \le 4$ constraint to the remaining three boxes.

Furthermore, a secondary 'tail-end' trap exists where students confuse the Negative Binomial requirement ('3rd success exactly on the 6th trial') with a standard Binomial probability ('exactly 3 successes in 6 trials'), leading them directly to the trap option 5/16.

Quick Facts About This Question

Question Information

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