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Question #40863

Difficulty: 8/10 | 🟢 Free

The Question

Let $\pi$ be a permutation of the set $\{1, 2, 3, 4\}$ chosen uniformly at random. Define indicator variables $I_k$ for $k \in \{1, 2, 3\}$ as follows: $I_k = \begin{cases} 1, & \text{if } \pi(k) > \pi(k+1) \\ 0, & \text{otherwise} \end{cases}$ Consider the region $R = \{(x,y) \in \mathbb{R}^2 : |x| \le 4, \text{ and } |y| \le 4\}$ . Let $A$ be the area of the locus of points $(x,y) \in R$ that satisfy the inequality: $I_1 x^2 + I_2 y^2 \le 4 I_3$ Find the expected value of $A$ , denoted as $E[A]$ .

Options

A.
$\frac{128 + \pi}{6}$
B.
$\frac{112 + \pi}{6}$
C.
$\frac{48 + \pi}{2}$
D.
$\frac{64 + \pi}{3}$

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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

Locus via Distance Constraints|Combinatorial Probability|Indicator Function Logic

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to handle locus constraints where the boundaries are dictated by indicator functions of dependent random variables.

The primary trap is the assumption of independence among the indicator variables $I_1, I_2, I_3$ .

Because they are defined on the same permutation, their probabilities are highly correlated (e.g., $P(I_1=1, I_3=1) = 6/24 \neq 1/4$ ).

A secondary trap lies in the boundary condition when $I_3 = 0$ .

Students often incorrectly assume that $0 x^2 + 0 y^2 \le 0$ implies an empty set (Area = 0) rather than the entire universal set (Area = 64), filtering out those who mechanically process inequalities without set-theoretic reasoning.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Locus via Distance Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free