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.)
Since each member must sit next to two members of the other party, the seating arrangement must be alternating between Democrat and Republican.
We have \(5!\) ways to seat the Democrats and \(5!\) ways to seat the Republicans. However, since we can rotate the seatings to match each other, we have to divide by \(5\).
Therefore, there are \(5!5!/4=2880\) different seating arrangements.