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

When people sit around a round table, rotations look the same. If everyone shifts one seat clockwise, it's still the same arrangement. For $n$ people at a round table, there are $(n-1)!$ arrangements (not $n!$).

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First, decide which people sit at which table: 7 people go to the 7-seater table and 8 people go to the 8-seater table.

Number of ways to choose 7 people from 15:

$\dfrac{15!}{7! \times 8!}$

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For the round table with 7 people, the number of arrangements is:

$(7-1)! = 6!$

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For the round table with 8 people, the number of arrangements is:

$(8-1)! = 7!$

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Total number of ways:

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

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

$= \dfrac{15! \times 6!}{8!}$

$= \dfrac{15!}{8!} \times 6!$

Therefore, the answer is $\dfrac{15!}{8!} \times 6!$