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Question #41848

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, \dots, 15\}$ . A triplet $(x, y, z)$ is chosen at random from the set $T = \{(a, b, c) \in S^3 \mid a \le b \le c\}$ . Let $E$ be the event that the triplet satisfies the conditions $y - x \ge |x - 7| \quad \text{and} \quad z - y \ge |y - 7|$ Find the probability $P(E)$ .

Options

A.
$\frac{37}{136}$
B.
$\frac{45}{136}$
C.
$\frac{35}{136}$
D.
$\frac{41}{136}$

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

Multiplication Principle Scope|Conditional Probability Constraints|Sample Space Partitioning|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to partition a sample space under constraints defined by a modulus function, specifically targeting the 'Modulus Sign Dependency' trap.

Students often attempt to solve inequalities like $y - x \ge |x - 7|$ by assuming a uniform linear relationship (e.g., $y \ge 2x - 7$ ), ignoring the piecewise nature of the modulus.

The 'Lethal Collision' occurs at the boundary $x=7$ and $y=7$ , where the multiplication factor for subsequent choices fundamentally shifts from a constant lower bound to a linearly increasing lower bound.

Failing to split the sample space into $x \le 7$ and $x > 7$ leads to severe undercounting or overcounting, punishing rote combinatorial formula application.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Multiplication Principle Scope
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free