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Question #41085

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ . A permutation of the elements of $S$ is chosen uniformly at random. Let $K$ be the set of all integers $x$ satisfying the inequality $|x^2 - 10x + 21| \le 3$ . What is the probability that for all $x \in K$ , the elements $x$ and $x+3$ are NOT adjacent in the chosen permutation?

Options

A.
$\frac{101}{252}$
B.
$\frac{134}{315}$
C.
$\frac{43}{105}$
D.
$\frac{17}{42}$

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

De Morgan’s Laws in Probability|String Method (Adjacency)|Modulus Inequality Logic

Why This Question Is Tricky

Common Mistakes

The question tests the application of De Morgan's Laws and the String Method under constrained conditions.

The primary trap lies in the modulus inequality $|x^2 - 10x + 21| \le 3$ , which yields the set $K = \{3, 4, 6, 7\}$ .

Students must identify the forbidden adjacent pairs: $(3,6), (4,7), (6,9), (7,10)$ .

The critical error occurs when students blindly apply the standard Inclusion-Exclusion formula assuming these pairs are disjoint (yielding the trap answer 134/315).

In reality, the pairs overlap at elements 6 and 7, requiring the formation of 3-element strings (e.g., 3-6-9) and significantly altering the intersection cardinalities in the Inclusion-Exclusion expansion.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	De Morgan’s Laws in Probability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free