You answer is definitely false.
just take n=1 as a counterexample for your solution.
Suppose there are families with people in each family for a total of people arranged around a circular table. How many ways are there to seat the people such that each person sits next to another member of his family?
So fix people from each family (treat them as a unit). This leaves ways to arrange the rest?
Hint:for a family, there is only two way to be seated:
(1)all the members are seated togather as a unit.
(2)all the members are divided into two group consisting of 2 and 3 members, members in each group are seated togather as a unit, and two units are seperated by members of other family.
No, It is possible that some fimalies are seperated into two Units(groups) by other family.
for example, here is a example of n=2 to clearify my posted thread:
(1)No family is seperated: that is all members in the same family are seated togather as a unit.
(2)For the case n=2, only one family are seperated is equivalent to (1). If
n>2, it is possible that one or some families are seperated by other family. We need a arguement on the number of families that are seperated.
(3)Two families are seperated, each family are divided into two groups consisting of 2 and 3 members.
Denote the two family members by .
Here is a example that all members are seated in a circle clockwisely:
similarly, You can find all the ways that all members are seated in a circle such that each member sits next to a member of his family.