A Senate committee has 5 Democrats, 5 Republicans, and 1 Independent.  In how many ways can they sit around a circular table if all the members of each party all sit next to each other?  (Two seatings are considered equivalent if one is a rotation of the other.)
I want to count the number of ways to arrange the 11 senators around a circular table, but I have some constraints and symmetries to deal with. First, let me fix the position of the Independent senator, say at the top of the table. This will eliminate any rotational equivalence, since the Independent will always be in the same spot. Now I have 10 seats left to fill, and I need to group the Democrats and Republicans together. There are 2 ways to choose which party sits to the left of the Independent, and which to the right. Then, there are 5! ways to arrange the Democrats in their group, and 5! ways to arrange the Republicans in their group. So, the total number of ways to sit the senators around the table is 2 * 5! * 5!, using the multiplication principle. 