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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.)

 May 23, 2021
 #1
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The number of ways is 2*4!*5! = 5760.

 May 23, 2021
 #2
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I'm sorry, but could you explain your answer?

viola  May 23, 2021
 #3
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Well, there actually is only one way, because if each person has to sit in-between the two people of the opposite party, then due to symmetry, there is, again, only one way to seat them.

DRDRDRDR 

(d=democrat, r = republican) Hope this helps! :) 

 May 23, 2021
 #4
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Right idea, but this means that every Republican is the same person and every Democrat is the same person.

 

If each member sits next to two members of the other party, then they should be alternating around the table. There is one way for this to happen: RDRDRDRDRD. I think that the first guest had the "2*" because he/she thought that this was different from DRDRDRDRDR. However, usually with table problems, it is the same rotation.

 

However, there are 5! ways to rearrange the Republicans and 5! ways to rearrange the Democrats. We multiply these terms because for every seating for the Republicans there is a seating for the Democrats and vice versa, so we get 5!*5!. For those keeping score at home, this is 14400.

 

 

Disclaimer: Although I usually feel good about these types of problems, this answer was not checked by any teacher or website and it is currently 11:43 pm where I am so please reply to this comment if I missed something or did something wrong. :)

Guest May 24, 2021

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