If 20 persons were invited for a party, in how many ways can they and the host be seated at a circular table?
In how many of these ways will two particular persons be seated on either side of the host?

Explanation

Clearly, there are 21 persons, to be seated round a circular table.
(i) Let us fix the seat of one person, say the host. The remaining 20 persons can now be arranged in 20! ways.
Hence, the number of ways in which these 21 persons can be seated round a circular table = 20!.
(ii) The two particular persons can be seated on either side of the host in 2 ways and for each way of their taking seats, the remaining 18 persons can be seated at the circular table in 18! ways.
`therefore` the number of ways of seating 21 persons at a circular table with two particular persons on either side of the host `=(2xx18!).`