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Question #40566

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be a set of distinct beads. $A$ and $B$ are subsets of $S$ such that $n(A) = 8$ , $n(B) = 6$ , and $n(A \cap B) = 2$ . All the beads that belong to exactly one of the sets $A$ or $B$ are selected. These selected beads are used to form two separate bracelets, each containing the same number of beads. Find the total number of distinct configurations of the pair of bracelets. (A configuration is invariant under the spatial rotation and flipping of individual bracelets, as well as the swapping of the two bracelets).

Options

A.
18144
B.
36288
C.
72576
D.
145152

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

Circular Permutations Constraint|Division into Groups|Difference vs. Symmetric Difference

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Set Theory (Symmetric Difference) and Combinatorics (Division into groups and Circular Permutations).

The primary trap is failing to divide by $2!$ when partitioning distinct objects into unlabelled groups of equal size, leading to the incorrect answer 36288.

A secondary trap is confusing bracelets (which possess reflectional symmetry, requiring division by 2) with standard circular permutations, which would lead to 72576.

Additionally, students might incorrectly calculate the number of selected beads by using the union or simple difference instead of the symmetric difference.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Circular Permutations Constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free