+0  
 
0
101
4
avatar

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.)

 Jun 14, 2022
 #1
avatar+162 
+1

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.

 Jun 15, 2022
 #2
avatar+124676 
+1

Seat  a Democrat  in any seat

 

Then we  have 4!  ways to seat  the other Democrats  and 5!  ways to seat the Republicans

 

So

 

4!  * 5!  = 

 

24 * 120  = 

 

2880  ways    { as RainbowSquirrel found !!! }

 

 

cool cool cool

 Jun 15, 2022

32 Online Users