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

First, let's fix one member's position. Without loss of generality, let's say a Democrat is seated at the 12 o'clock position.

Now, we need to arrange the remaining 9 members around the table such that each member sits next to two members of the other party. Since there are 5 Democrats and 5 Republicans, they need to alternate.

The possible seating arrangement starting from the Democrat at 12 o'clock position can be represented as: D-R-D-R-D-R-D-R-D-R

There are 5! ways to arrange the Democrats and 5! ways to arrange the Republicans within their own groups.

Therefore, the total number of ways the committee members can sit around the circular table is 5! * 5! = 120 * 120 = 14, 400 ways.