In how many ways can 15 people be seated around two round tables with seating capacities of 7 and 8 people?

'n' objects can be arranged around a circle in (n - 1)! ways.
If arranging these 'n' objects clockwise or counter clockwise means one and the same, then the number arrangements will be half that number.
i.e., number of arrangements = $$\frac{{\left( {n - 1} \right)!}}{2}$$
You can choose the 7 people to sit in the first table in ¹⁵C₇ ways.
After selecting 7 people for the table that can seat 7 people, they can be seated in:
(7 - 1)! = 6!
The remaining 8 people can be made to sit around the second circular table in: (8 - 1)! = 7! Ways.
Hence, total number of ways: ¹⁵C₇ × 6! × 7!