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Question #40334

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{-3, -2, -1, 0, 1, 2, 3\}$ . A relation $R$ on $S$ is defined such that it represents a bijective function $f: S \to S$ (i.e., $R = \{(x, f(x)) \mid x \in S\}$ ). Find the total number of such relations $R$ where no element $(x, y) \in R$ satisfies $y = x$ or $y = -x$ .

Options

A.
672
B.
720
C.
1152
D.
1854

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

Total Number of Relations|Derangements (Inclusion-Exclusion)|Geometry-based Combinations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map geometric constraints on relations to advanced combinatorics (restricted permutations).

The trap lies in treating the conditions $f(x) \neq x$ and $f(x) \neq -x$ as standard derangements or assuming the cycles are independent without accounting for overlapping domains.

By disguising a 'Problème des ménages' variant behind Cartesian line equations, it forces the student to partition the forbidden positions into disjoint sub-boards and correctly apply the Inclusion-Exclusion Principle or Rook Polynomials.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Total Number of Relations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free