Question

In how many ways can n men be seated around a round table when in no two ways a man has the same neighbours?

Answer 1

For seating on a round table we have to fix up, one man at a place then there are n -1 persons left which can be arranged in (n -1)! ways. Now suppose Mr. Khanna is seated first and in any arrangement, he has Mr. Lai and Mr. Sehgal on right and left or on left and right. In the two ways, Mr. Khanna has the same neighbours.
Hence the required number. of: arrangements is$\frac{1}{2}$ (n-1)! because in no two ways a person has the same neighbours.