Results 1 to 4 of 4

Math Help - couples

  1. #1
    asc
    asc is offline
    Newbie
    Joined
    Dec 2007
    Posts
    8

    couples

    How many ways are there to seat n married couples at a round table with 2n chairs in such a way that the couples never sit next to each other?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member wingless's Avatar
    Joined
    Dec 2007
    From
    Istanbul
    Posts
    585
    Edit: You may ignore this incorrect solution, I leave it for people to see where I did wrong.

    --------------------

    Firstly, we will find the number of the ways that the couples sit together.


    As there are n couples, there will be (n-1)! different arrangements for the couples. And each couple can sit in 2! ways.



    If a couple can sit in 2 ways, n couples can sit in 2^n ways.

    Hence, n couples can be placed in 2^n (n-1)! ways. (when every couple sits together)

    All arrangements - Couples sitting together = Couples not sitting together

    We can place 2n people in (2n-1)! ways.

    So, the answer is
    All arrangements - Couples sitting together

    (2n-1)! - 2^n (n-1)!
    Last edited by wingless; January 10th 2008 at 11:02 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by wingless View Post
    \text{All arrangements}-\text{Couples sitting together}=\text{Couples not sitting together}
    It does not quite work that way.
    The complement of “all couples together” is not “no couples together”.

    What you want here is “no couples together” which is the complement of “at least one couple is together” .
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member wingless's Avatar
    Joined
    Dec 2007
    From
    Istanbul
    Posts
    585
    Quote Originally Posted by Plato View Post
    It does not quite work that way.
    The complement of “all couples together” is not “no couples together”.

    What you want here is “no couples together” which is the complement of “at least one couple is together” .
    Oh, you're quite right! That was such a big, silly mistake.. So my solution is totally wrong
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. couples problem
    Posted in the Statistics Forum
    Replies: 4
    Last Post: January 20th 2011, 03:59 PM
  2. couples problem
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: January 13th 2011, 09:13 AM
  3. Arrangement of couples
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: October 17th 2009, 05:58 AM
  4. Married couples in a row
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: September 21st 2009, 04:33 AM
  5. Given twenty couples...
    Posted in the Math Topics Forum
    Replies: 3
    Last Post: February 1st 2007, 07:12 AM

Search Tags


/mathhelpforum @mathhelpforum