1. ## circular permutation

Suppose there are $n$ families with $5$ people in each family for a total of $5n$ 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 $2$ people from each family (treat them as a unit). This leaves $(5n-2n)! = 3n!$ ways to arrange the rest?

2. You answer is definitely false.
just take n=1 as a counterexample for your solution.

3. 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.

4. Originally Posted by Shanks
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.
So seat one family and there are $(n-1)!$ ways of arranging the other members.

5. 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 $A_1,...,A_5; B_1,...,B_5$.
Here is a example that all members are seated in a circle clockwisely:
$A_1,A_2,B_1,B_2,A_3,A_4,A_5,B_3,B_4,B_5$
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.