1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Counting Seating Arrangments of Couples at a Round Table

  1. Sep 12, 2007 #1
    [SOLVED] Counting Seating Arrangments of Couples at a Round Table

    I'm reading this example in my probability book which is I'm not understanding. It says:

    There are 19! ways of arranging 20 people around a table. The number of arrangements that result in a specified set of n men sitting next to their wives can most easily be obtained by first thinking of each of the n married couples as being single entities. If this were the case, then we would need to arrange 20 - n entities around a round table, and there are clearly (20 - n - 1)! such arrangements.

    There are 10 married couples by the way. The "20 - n entities" part is bugging me. Shouldn't that be 10 - n, given that there are 10 entities/married couples. I also don't understand how the (20 - n - 1)! part follows.
     
  2. jcsd
  3. Sep 12, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Why ist it 10-n? If there is one married couple, there are 18 singletons, hence 19 objects to arrange (amazing what thinking of an example can do..). Thus if there are n couples hence how many single people? Now how many 'objects' are you arranging in a circle?
     
  4. Sep 12, 2007 #3
    Technically, there are no single people (they're all in couples). However, if n of the couples have already been seated, then there are 20 - 2n seats around the table for the remaining 10 - n couples or 20 - 2n people.
     
    Last edited: Sep 12, 2007
  5. Sep 12, 2007 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    What does that show? (Apart from the fact that you seem to be focussed on the wrong thing.)
     
  6. Sep 12, 2007 #5
    To be honest, I just don't understand the explanation. Here's how I would count the seating arrangements:

    First, I would pick an ordering of the pairs and number them 1 through 20. Then I would sit pair 1 by first picking a seat for the woman and then picking a side for the man. This can be done in one of 20 * 2 ways.

    Then I proceed with pair 2. The woman can sit in one of 18 ways. In 2 of those ways, the man is forced to sit in one spot. For the other 16 locations, the man can sit to the right or left. Hence, there are 2 + 16 * 2 ways to sit pair 2.

    For pair 3, things get someone more complicated because I have to take into account of where pairs 1 and 2 are sitting. Ditto for pairs 4 - 10.
     
  7. Sep 13, 2007 #6

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Forget the married status or otherwise of the objects.

    1. There are 20 objects,

    2. we pair up 2n of them in n pairs.

    3. We wish to arrange these n pairs and 20-2n remaining unpaired items in a circle.

    4.That is we have 20-2n+n=20-n things to put in a circle

    5. which can be done (20-n-1)! ways.

    which of 1-5 is confusing?

    Notice that the question does not distinguish between the order of the two objects in a pair, just that they are paired.
     
  8. Sep 13, 2007 #7
    OK. I understand now. 3 had me confused because I thought it didn't make any sense to arrange paired and unpaired objects. Thank you.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Counting Seating Arrangments of Couples at a Round Table
Loading...