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Question #41112

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_5$ be the set of all permutations of the set $\{1, 2, 3, 4, 5\}$ . An equivalence relation $\sim$ is defined on $S_5$ such that for $\sigma, \tau \in S_5$ , $\sigma \sim \tau$ if and only if $\sigma$ and $\tau$ have the exact same number of fixed points. A permutation $\sigma$ is chosen uniformly at random from $S_5$ . Let $[\sigma]$ denote the equivalence class containing $\sigma$ . What is the expected value of $|[\sigma]|$ , the cardinality of the equivalence class containing $\sigma$ ?

Options

A.
$\frac{2231}{60}$
B.
24
C.
20
D.
$\frac{2231}{120}$

Check Your Answer

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

Equivalence Class Partitioning|Sample Space Partitioning|Equiprobable Outcomes|Equiprobability Fallacy

Why This Question Is Tricky

Common Mistakes

This question targets the 'Equiprobability Fallacy' within sample space partitioning.

Students often incorrectly assume that the expected value of the class size is simply the arithmetic mean of the distinct class sizes (yielding $120/5 = 24$ or $120/6 = 20$ ).

However, because a permutation is chosen uniformly at random, larger equivalence classes have a proportionally higher probability of being selected.

The correct expectation requires weighting each class size by its own cardinality, effectively computing the second moment of the partition sizes divided by the total sample space.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Equivalence Class Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free