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.)
+128707 "Anchor' the Democrats in any 5 consecutive seats. And there are 5! = 120 ways to arrange them.
Next, the Republicans can either be on the left or right of the Dems......so that's 2 ways of choosing where they sit. And they can be individually arranged in 5! ways, too = 120 ways.
The independent gets the remaining chair by default.....so we have
5! x 2 x 5! = 120^2 x 2 = 28,800 possible arrangements
Another way to do this is to "anchor" the independent first.......the same logic would still apply.....
+128707 "Anchor' the Democrats in any 5 consecutive seats. And there are 5! = 120 ways to arrange them.
Next, the Republicans can either be on the left or right of the Dems......so that's 2 ways of choosing where they sit. And they can be individually arranged in 5! ways, too = 120 ways.
The independent gets the remaining chair by default.....so we have
5! x 2 x 5! = 120^2 x 2 = 28,800 possible arrangements
Another way to do this is to "anchor" the independent first.......the same logic would still apply.....
