Question

2n persons are to be seated n on each side of a long table r (<n) particular persons desire to sit on one side; and s (<n) other persons desire to sit on the other side. In how many ways can the persons be seated?

Answer 1

For the side where r persons desire to sit, we need (n – r) more persons. This (n – r) may be chosen from (2n – r – s) in ^{(2n - r - s)} C_{n - r} - - - ways. Automatically, the remaining (n – s) persons go to the other side where already there are s desirous of seating. Thus, there are ^{(2n - r - s)}C_{n - r} - - - ways of distributing n persons for each side provided with the restriction of r on one side and s on the other side. n persons on each side can be permuted in n seats in n! ways. The number of ways of seating the 2n persons, n on each side, is therefore, ^{(2n - r - s)}C_{n - r}(n!)².