Fifteen people are to be seated around two round tables with capacities 7 and 8. Assuming the two tables are distinguishable by size, in how many ways can they be seated (rotations at each table considered identical)?

Introduction / Context:
We split people between two round tables of different sizes and then account for circular symmetry at each table. Since capacities differ (7 vs 8), the tables are distinguishable by size.

Given Data / Assumptions:

• Total people = 15.
• Table A seats 7, Table B seats 8.
• Circular seatings: rotations identical at each table; reflections distinct (standard seating convention).

Concept / Approach:
First choose which 7 people sit at the 7-seat table: C(15,7) = C(15,8). Then arrange those 7 around a circle: (7−1)! = 6!. Arrange the remaining 8 around the other circle: (8−1)! = 7!.

Step-by-Step Solution:
Distribute people: C(15,7) (equivalently C(15,8)).Seat at 7-table: 6! arrangements.Seat at 8-table: 7! arrangements.Total = C(15,8) * 6! * 7!.

Verification / Alternative check:
Because table sizes differ, swapping entire tables cannot overcount; thus no division by 2 is needed.

Why Other Options Are Wrong:
15!/(8!) is dimensionally inconsistent; 15C8*8! ignores circular reduction; the “7!/88!” option is malformed.

Common Pitfalls:
Dividing by 2 as if tables were identical when capacities already distinguish them.

Final Answer:
15C8 x 6! x 7!