Round-table seating with a restriction: Twelve persons are to be seated around a circular table. Two particular persons must not sit side by side. How many distinct circular arrangements are possible?

Introduction / Context:
Circular permutations differ from linear ones because rotations are indistinguishable. This problem asks for the number of round-table arrangements of 12 people with a specific adjacency restriction on two identified persons.

Given Data / Assumptions:

• Total persons = 12.
• Two particular persons must not be adjacent.
• All individuals are distinct; reflections are considered different unless stated otherwise (standard round-table convention counts rotations as the same, not reflections).

Concept / Approach:
Use “Total − Restricted.” First compute all circular arrangements without restrictions, then subtract the arrangements where the two specified people sit together as a block. For circular arrangements of n distinct people, the count is (n − 1)!.

Step-by-Step Solution:

Verification / Alternative check:
Anchor one person to eliminate rotation; counting linearly then adjusting leads to the same 9 * 10! figure.

Why Other Options Are Wrong:

• 2 (10!) counts only adjacent cases.
• 10! omits most permissible configurations.
• 45 (8!) is unrelated to circular-block logic here.

Common Pitfalls:
Forgetting the factor 2 for the pair’s internal order, or using 12! instead of 11! for circular arrangements.

Final Answer:
9 (10!)