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Question #40580

Difficulty: 8/10 | 🟢 Free

The Question

A system distributes $n$ identical tasks to 3 distinct servers. The true probability that no server is idle is $P$ , assuming each task is assigned independently and uniformly at random to one of the servers. A flawed model assumes that all possible valid configurations (non-negative integer distributions of the $n$ tasks among the 3 servers) are equally likely, yielding a probability $Q$ that no server is idle. If $\frac{P}{Q} = \frac{56}{27}$ , what is the value of $n$ ?

Options

A.
5
B.
6
C.
7
D.
8

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

Distributing Identical Items (Stars and Bars)|Complementary Counting|Equiprobability Fallacy

Why This Question Is Tricky

Common Mistakes

This problem targets the Equiprobability Fallacy, a classic trap in combinatorial probability.

Students often confuse the sample space of independent events (Maxwell-Boltzmann statistics, where each of the $3^n$ outcomes is equally likely) with the sample space of indistinguishable configurations (Bose-Einstein statistics, where each of the $\binom{n+2}{2}$ Stars-and-Bars outcomes is incorrectly assumed to be equally likely).

By forcing the calculation of both the correct independent-assignment probability (requiring Inclusion-Exclusion) and the flawed configuration-based probability, the question explicitly tests the student's ability to distinguish between physical identicalness and probabilistic independence.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Distributing Identical Items (Stars and Bars)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free