Before You Begin

Question #41893

Difficulty: 8/10 | 🟢 Free

The Question

Let $M$ be the multiset $\{1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6\}$ . A circular arrangement of $M$ is defined as an equivalence class of linear permutations of $M$ under cyclic shifts. Let $S$ be the set of all circular arrangements of $M$ that satisfy both of the following conditions: (I) The arrangement has at least one axis of reflectional symmetry. (II) No two identical numbers are adjacent. Find the number of elements in $S$ .

Options

A.
360
B.
720
C.
1080
D.
1440

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

Indicator Function Logic|String Method (Adjacency)|Symmetry vs. Antisymmetry

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between symmetry and adjacency constraints (the 'antisymmetry trap').

Students often attempt to use Burnside's Lemma or count all symmetries without analyzing the geometric implications of the constraints.

The critical trap is failing to recognize that an Edge-Edge (E-E) reflection inherently forces adjacent elements to be identical, which is strictly forbidden by the adjacency constraint.

By blindly counting all symmetries, students overcount by including these invalid E-E symmetric cases (leading to 720 or 1440).

Recognizing that only Vertex-Vertex (V-V) symmetries are valid bypasses complex inclusion-exclusion.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Indicator Function Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free