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Question #41049

Difficulty: 8/10 | 🟢 Free

The Question

Four married couples (8 distinct people) are to be seated around a circular table. Let $X$ be the number of arrangements in 8 distinct, numbered chairs such that no couple sits diametrically opposite to each other. Let $Y$ be the number of arrangements in 8 identical, unnumbered chairs (where arrangements are identical if they differ only by rotation) such that exactly one couple sits diametrically opposite to each other. Find the value of $\frac{X}{Y}$ .

Options

A.
15
B.
$\frac{15}{8}$
C.
120
D.
$\frac{15}{64}$

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

Algebra of Events|Combinatorial Counting (P&C Integration)|Circular Permutations Constraint

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the Algebra of Events (via the Principle of Inclusion-Exclusion and De Morgan's Laws) with circular permutations.

The core trap lies in the distinction between 'labeled' (numbered) and 'relative' (unnumbered) circular spaces.

Students frequently ignore the symmetry-breaking effect of numbered chairs, assuming the probability or relative counts apply directly without scaling.

By asking for a ratio of counts from two differently defined sample spaces ( $X$ in a fixed space, $Y$ in a rotational space), the problem forces the student to explicitly handle the factor of 8.

A failure to distinguish these spaces leads to the trap answer 15/8, punishing the over-reliance on relative permutation formulas.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Algebra of Events
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free