There are 5 clients and 5 consultants at a round table meeting. In how many…

Concept: When two groups must be seated around a circular table so that no two members of one group are adjacent, first seat the unrestricted group in a circle — this gives (n − 1)! arrangements — and then place the restricted group into the gaps between them (n gaps are created for n people already seated), giving a further n! arrangements. This is the standard gap method for circular arrangements with an adjacency restriction.

1. There are 5 clients. Since rotations of a circular arrangement are equivalent, fixing one client's position and permuting the remaining 4 clients gives (5 − 1)! = 4! = 24 ways to seat the clients.

2. Seating the 5 clients around the table creates exactly 5 gaps, one between each pair of consecutive clients.

3. Since no two consultants may sit together, at most one consultant can go in each gap. With exactly 5 consultants and 5 gaps, the 5 consultants can be arranged into these gaps in 5! = 120 ways.

4. Total number of arrangements = 4! × 5! = 24 × 120 = 2880.

Cross-check: With no adjacency restriction at all, all 10 people could be seated in 9! = 362,880 ways. The restricted count of 2880 is far smaller, consistent with the adjacency rule eliminating the vast majority of arrangements in which some consultants would sit together.