Question

20 people are invited for a party. The different number of ways in which they can be seated at a circular table with two particular person seated on either side of the host is:
A) 19!2!
B) 18!2!
C) 20!2!
D) 18!3!

Answer 1

Hint- The number of people given in the question is 20. We will exclude the host of the party out of the total number and then take the possibility of the two persons sitting on either side of the host to interchange their seats.

Complete step-by-step answer:
Now, the number of persons given in the question excluding the host is 20.
Since, two persons are particularly seated on either side of the host so we will subtract two from the total persons in the party i.e. 20. We will add those two persons with the host and make a group which will leave us with total number of people-
18 people + 1 group = 19
Finding out the number of arrangements for people sitting on a circular table = (n-1)!, where n is equal to the total number of people.
Thus, the arrangement will be (19-1)! = 18!
Number of arrangements of people sitting on the circular table= 18!
Now, we will find out the arrangement of those two persons sitting on either side of the host, we will get-
Arrangement of 2 persons sitting on either side of the host = 2! (As they can interchange their seats)
Thus, we will get our answer by finding out the total number of arrangements i.e.-
Total number of arrangements of 20 persons = 18!2!
Thus, B is the correct answer.

Note: Do not forget to take the possibility of the two particular persons sitting on either side of the host to change their seats as it is not given in the question that they do not change their seats at all. If you do forget, you will get the wrong answer.