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.)

ant101
Nov 4, 2018

#1**+1 **

If each member sits next to two members of the other party, they must just alternate D - R - D - R, etc

So...."anchor" one member, say, a Republican, in any position

Going counter-clockwise [ we could also go in the other direction ], we have

5 choices for the next seat [ one of 5 Dems ], 4 for the next [ one of 4 remaining Repubs ], 4 for the next [ one of the 4 remaining Dems, 3 for the next [ one of the 3 remaining Repubs ]....etc.

So...the number of possible arrangements = 5! * 4! = 120 *24 = 2880

CPhill
Nov 4, 2018

#2**+1 **

Hmmmm.....isn't it 5! x 5!

5 choices for first seat 5 choices second seat 4 choices third seat etc etc

5! x 5!

I always seem to get these quite wrong! ~EP

ElectricPavlov
Nov 4, 2018

#3**+2 **

Notice, EP, that all rotations of any arrangement are the same.....so.....a Repub assigned to any one seat can be in this same seat no matter the rotation....the only difference in the arrangements are the number of ways that the other people can be seated around him [ keeping in mind that we must have alternating seating patterns of R - D - R - D, etc ]

To see this more easily....in a simplified example....suppose that we have 4 people seated at a table

One arrangement is

1

4 2

3

This is the same arrangement [ with respect to rotations ] as

4

3 1

2

So.....if we anchor the first person at the " top "...the only difference in arrangements [ with respect to rotations ] are the ways the other three people are seated around him.....note that another possible arrangement is

1

3 4

2

CPhill
Nov 4, 2018