Question

There are 20 persons including two brothers. In how many ways can they be arranged on a round table if:
There is exactly one person between the two brothers.

Answer 1

Let the two brothers be taken out and the remaining 18 persons can be arranged on a circular table in (17!) ways, Now there are 18 persons on the table. Two brothers can be arranged about each of them on either side in 2! = 2 ways because there is to be exactly one person between the two brothers. Hence the required number by fundamental
theorem is
(17) ! $\cdot$ (18.2) = 2 (18) !