In a G 20 meeting there are 20 different representatives sitting around a circular table.\nIn how many different ways can they be arranged if there must be exactly one person sitting between the representatives Manmohan and Musharraf?

Introduction / Context:
This question focuses on circular permutations with a specific spacing condition between two distinguished people. In circular arrangements, rotations are considered the same, so we use special counting rules. The requirement that exactly one person sits between Manmohan and Musharraf introduces a relative positioning constraint that must be carefully handled.

Given Data / Assumptions:

• Total representatives = 20, all distinct people.
• Two special representatives: Manmohan and Musharraf.
• They sit around a circular table where rotations of the same order are considered identical.
• Exactly one person must sit between Manmohan and Musharraf on both sides, meaning their seats are separated by exactly one chair.
• All remaining representatives may sit in any order subject to this condition.

Concept / Approach:
For circular permutations with constraints, a common method is to fix the position of one distinguished person to break rotational symmetry. Once we fix Manmohan's seat, we place Musharraf in a seat that is exactly two positions away (one seat between them). There are exactly two such seats: one two places clockwise and one two places anticlockwise. After fixing both special people, we can freely permute the remaining people in the remaining seats. The count of these free permutations gives the final answer.

Step-by-Step Solution:
Fix Manmohan at a reference seat to remove circular symmetry. This gives 1 effective position for him. Identify the seats where Musharraf can sit so that there is exactly one person between them. There are two such positions: two seats clockwise or two seats anticlockwise from Manmohan. Therefore Musharraf has 2 valid choices once Manmohan's seat is fixed. The remaining representatives = 20 - 2 = 18 people. These 18 people can be seated in the remaining 18 seats in 18! ways. Total valid arrangements = 2 * 18!.

Verification / Alternative check:
Another way to think about this is to first imagine placing Manmohan and Musharraf with exactly one seat between them, treating the pair as a structured configuration on the circle. Once Manmohan is fixed, any seating that keeps Musharraf two seats away is determined by which side he takes. Because there are always exactly two choices around a circle for such a spacing, and because each of those can be extended by arranging the remaining 18 people in any order, the total of 2 * 18! is consistent. There is no double counting because each final seating corresponds to exactly one choice of reference position for Manmohan.

Why Other Options Are Wrong:

• 2 * 17!: This ignores one factor of 18 and undercounts, as it assumes only 17 free seats remain instead of 18.
• 3! * 18!: This adds an unjustified factor of 3!, overcounting arrangements.
• 17!: This assumes only 18 people are being arranged in a circle, which is not the scenario here.

Common Pitfalls:
A common mistake is to treat circular arrangements as linear, leading to an extra factor of 20 for rotations. Another frequent error is miscounting how many seats place Musharraf at the correct distance from Manmohan. Some learners also forget to reduce the circular symmetry by fixing one person first and attempt to handle the rotation at the end, which can cause double counting. Fixing one person and then working with simple factorial logic is usually the safest route.

Final Answer:
The number of circular arrangements with exactly one person between Manmohan and Musharraf is 2 * 18!.