Say there are n boys and n girls seated in a circle with boys and girls alternating. What will be the probability that no girl is sitting beside her brother and no boy is sitting beside his sister. Each boy has only one sister and each girl has only one brother.
The no of ways $2n$ boys and girls can seat in the circle=$(2n-1)$!
As from the question,there are $n$ pairs of siblings.
The no. of way $n$ 'sibling couple' can be seated in the circle=$(n-1)$!
Each sibling can also sit together in $2$ different ways
The no. of ways we can make siblings sit together is=$2(n-1)$!
The no. of ways we cannot make siblings sit together is=$(2n-1)!-2(n-1)$!