A gentleman invites 13 guests to a dinner and places 8 of them at one table and remaining 5 at the other, the tables being round. The number of ways he can arrange the guests is

The number of ways in which 13 guests may be divided into groups of 8 and 5 = 13C5=13!5!8!Now, corresponding to one such group, the 8 guests may be seated at one round table in (8 – 1)! i.e., 7! ways and the five guests at the other table in (5 – 1)! i.e., 4! ways.But each way of arranging the first group of 8 persons can be associated with each way of arranging the second group of 5, therefore, the two processes can be performed together in 7! × 4! ways.Hence, required number of arrangements=13!5!8!×7!×4!=13!5⋅4!8⋅7!×7!×4!=13!40