Question

A Senate committee has 5 Democrats and 5 Republicans. In how many ways can they sit around a circular table if each member sits next to two members of the other party? (Two seatings are the same if one is a rotation of the other.)

Answer 1

Step-by-step explanation:

Here's my take

Anchor the Republicans in any 5 seats.....note that it doesn't matter which 5 seats they occupy....if rotated, all of the arrangements will look the same......and they can be arranged in 5! = 120 ways

So.....the Democrats occupy the other 5 seats and they can be arranged in the same muber of ways = 5! = 120 ways

So...the total arrangemets are (Ways to arrange the Republicans) x ( Ways to arrange the Democrats) = 120 x 120 = 14,400

To see this..let's suppose that we have A,B,C,D and AB must sit as a block and CD must sit as a block

So we have 2! = 2 ways to arrange AB and 2! = 2 ways to arrange CD = 2! x 2! = 4 total arrangements

So.....the arrangements are

A A B B

C B D B C A D A

D C D C

Note that any rotations will still look the same with regards to order