# Circular permutation of 20 person and 1 host.

20 were invited for a party, find the no ways in which 20 persons and the host can be seated around a circular table such that two particular person be seated on either side of host.

Solution :

Since the place of host is fixed and the 2 particular person around him is fixed. Treating 2 person and 1 host as single unit.

No of ways = $(19 - 1)! * 2$ , but the solution does not match the answer provided with textbook.

Since we are treating the host and the $2$ special people sitting next to the host as a single unit, and since we are fixing this unit, it remains to permute the remaining $20-2=18$ party guests. Since the order in which the $2$ special people are sitting next to the host does not matter, we must multiply by $2$. Thus, we obtain: $$2\cdot18!$$ which is equivalent to your answer. What was the answer in the textbook?
• the given answer is 19! – AppDeveloper Jun 23 '13 at 5:46