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

$\frac{15!}{8!}×6!$

The correct answer is Option (4) → $\frac{15!}{8!}×6!$

Let the two round tables be distinct, with seating capacities 7 and 8.

Step 1: Choose people

• Choose 7 people out of 15 for the 7-seater table:

$\begin{pmatrix}15\\7\end{pmatrix}$

Step 2: Arrange them around round tables

• 7 people around a round table: $(7-1)! = 6!$
• 8 people around a round table: $(8-1)! = 7!$

Total number of ways

$\begin{pmatrix}15\\7\end{pmatrix} \times 6! \times 7!$

$= \frac{15!}{7!8!} \times 6! \times 7! = \frac{15! \times 6!}{8!}$