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Homework Help: 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

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    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

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    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

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    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.
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