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