Question

If \[20\] persons are invited to a party. In how many different ways can they and the host be seated at a circular table if the two particular persons are to be seated either side of the host?

Answer 1

Hint: Firstly we will find out in how many different ways can they and the host be seated at a circular table and secondly we have to find if the two particular persons are to be seated either side of the host after that find out the total ways where two particular persons are to be seated either side of the host and we can find out the required result.

Complete step-by-step answer:
Now, according to the given question:
Firstly we have to find in how many different ways can they and the host be seated at a circular table
Hence the total number of persons are \[20+1=21\]
The number of ways to be sorted in circular table \[=(n-1)!\] where \[n=21\]
 \[\Rightarrow (21-1)!\]
 \[\Rightarrow 20!\]
Secondly we have to find conditions for the two particular persons to be seated at either side of the host.
 As in the side of host two person will be seated in two ways:
Hence the number of remaining person will be:
 \[\Rightarrow (20-2)!\]
 \[\Rightarrow (18)!\]
Hence the total ways will be:
 \[\Rightarrow (18)!\times 2\]

Note: In probability Null set \[\phi \] and sample space \[S\] also represent events because both are the subsets of \[S\] . Here \[\phi \] represents an impossible event and \[S\] represents a definite event. The subset \[\phi \] of \[S\] denotes an impossible event and the subset of \[S\] of \[S\] itself denotes the sure event.