In how many ways can 15 people be seated around two round tables with seating capacities of 7 and 8 people?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{15!}{8!}\times6!\)
Around a round table only the order of the people matters, not where the circle starts, so \(n\) people can sit round a table in \((n-1)!\) ways. First the 7 people for the smaller table are chosen; the other 8 go to the larger table, and each table is then arranged.
Choosing the 7 people: \(\displaystyle {}^{15}C_{7}=\frac{15!}{7!\,8!}\)
Seating them round the table of 7: \((7-1)!=6!\)
Seating the other 8 round the table of 8: \((8-1)!=7!\)
Total \(\displaystyle =\frac{15!}{7!\,8!}\times6!\times7!=\frac{15!}{8!}\times6!\)