Family of four and eight people ordered around a table So the question is how many ways can a family of four which includes the mother, father and two children, be ordered around a table with eight other people if the mother must sit beside the father and there needs to be a child on either side. So for example: $S M F D$ 
is son, mother, father, daughter which is different from $S F M D$. Also rotation does not count as a new way. 
I have found that there are 4 ways to order the family, then I believe that there are $8!$ ways to order the other 8 people, and also we multiply by 3 since there are $3 * 4$ seats in total that the family can choose from. So the final answer I got way $4 * 3 * 8!$ but I dont feel right about it. Can someone help?
 A: To begin, group the family together as a single unit.  You have eight other people (I'll label as $x_1,x_2,x_3,\dots,x_8$) and the family (I'll label as $X$).
How many ways can you arrange the elements $x_1,x_2,\dots,x_8,X$ around a circle?

 pick one person to be special as a point of reference.  Seat everyone else around that person.  There are thus $8!$ number of ways.  By a similar argument, you could arrange the nine and then divide by symmetry to get $\frac{9!}{9}=8!$ number of ways.

Now that we have comfortably arranged the other guests around the family, we will decide how the family spreads out amongst themselves.  For the time being, we will treat the mother and father as one collective unit.  Let $S,D,P$ represent the son, daughter, and parents respectively.  How many ways are there to arrange $S,D,P$ in a line where the parents must take the middle spot and the children to either side.

 The location of $P$ is predetermined.  Thus, the only question is if the son is on the left or right, and the daughter takes the other space.  Thus: two possibilities.

Now, expand the Mother and Father to be separate entities.  Is the mother on the left or the right?

 Two possibilities

For a combined total of:

 $8!\cdot 2\cdot 2$

A: What is your reasoning for multiplying by 3?  You should consider the family as a single entity.  That gives you 9 entities to place around the table.  Use the table seating rules for figuring out that amount.
Then multiply that amount by the number of ways you determined the family can be arranged within its own group.
