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?
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A9 (10!)
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B2 (10!)
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C45 (8!)
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D10!
Answer
Correct Answer: 9 (10!)
Explanation
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:
Total circular arrangements = (12 − 1)! = 11!Treat the two as a single block + the other 10 ⇒ 11 items on a circleArrangements with the pair adjacent = (11 − 1)! * 2 = 10! * 2 (factor 2 for swapping within the pair)Required = 11! − 2 * 10! = (11 − 2) * 10! = 9 * 10!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!)