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Question #40583

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n$ be the set of all permutations of the set $\{1, 2, \dots, n\}$ where $n \ge 4$ . A permutation $\pi$ is chosen uniformly at random from $S_n$ . Let $X$ denote the number of fixed points of $\pi$ (i.e., the number of integers $i$ such that $\pi(i) = i$ ). What is the expected value of $X^3$ , denoted by $E(X^3)$ ?

Options

A.
$1$
B.
$3$
C.
$5$
D.
$8$

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

Derangements (Inclusion-Exclusion)|Random Variable Expectation|Equiprobability Fallacy

Why This Question Is Tricky

Common Mistakes

The problem tests the application of the Linearity of Expectation on dependent indicator variables.

A common trap is attempting to compute the expectation using the probability mass function of derangements, $P(X=x) = \frac{1}{x!} \sum_{k=0}^{n-x} \frac{(-1)^k}{k!}$ , which leads to an intractable algebraic summation under time constraints.

Another trap is the equiprobability fallacy, where students incorrectly assume the indicator variables are independent and multiply their individual expectations.

By expanding $X^3$ and applying linearity, the dependency structure is elegantly bypassed, revealing that the higher moments of fixed points match the Bell numbers.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Derangements (Inclusion-Exclusion)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free