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Question
A party has 20 participants and a host. Find the number of distinct ways for the host to sit with them around a circular table. How many of these ways have two specified persons on either side of the host?
Solution
A party has 20 participants.
All of them and the host (i.e., 21 persons) can be seated at a circular table in (21 1)! = 20! ways.
When two particular participants are seated on either side of the host.
Host takes the chair in 1 way.
These 2 persons can sit on either side of the host in 2! ways
Once the host occupies his chair, it is not a circular permutation any more.
Remaining 18 people occupy their chairs in 18! ways.
∴ Total number of arrangements possible if two particular participants are seated on either side of the host = 2! × 18!
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