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Math Help - Cross checking a simple permutation problem

  1. #1
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    Cross checking a simple permutation problem

    Hi guys,

    Q) In how many different ways can 6 gentlemen and 5 ladies sit around a table if no two ladies sit side by side?

    My answer: (6-1)! * 5! = 5! * 5!

    Am i correct?
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  2. #2
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    Hello, saberteeth!

    In how many different ways can 6 men and 5 women
    sit around a table if no two ladies sit side by side?

    Consider 12 chairs around the table.

    Seat the six men in alternate chairs.
    . . There are: . (6-1)! \:=\:120 ways.

    Now seat the five women in five of the six empty chairs.
    . . There are: . _6P_5 \:=\:720 ways.


    Therefore, there are: . 120\cdot720 \:=\:86,\!400 seating arrangements.

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, saberteeth!


    Consider 12 chairs around the table.

    Seat the six men in alternate chairs.
    . . There are: . (6-1)! \:=\:120 ways.

    Now seat the five women in five of the six empty chairs.
    . . There are: . _6P_5 \:=\:720 ways.


    Therefore, there are: . 120\cdot720 \:=\:86,\!400 seating arrangements.

    Hi Soroban,

    Thanks for your reply. I am confused.. Wouldn't that be the same permutation for 6 men and "6 women"? I read it on the 2nd example on this website: All about Circular Permutations | TutorVista.com
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  4. #4
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    Hello, saberteeth!

    Wouldn't that be the same permutation for 6 men and 6 women?
    Yes, it is!

    Look at it this way . . .

    There are 12 chairs.
    Once we seat the 6 men and 5 women . . . there is one empty seat.

    And that is where the 6th woman can sit.



    A similar example:

    There are six chairs in a row.
    In how many ways can five men be seated?

    The answer is a permutation: . _6P_5 \:=\:720 ways.


    There are 6 chairs in a row.
    In how many ways can six men be seated?

    The answer is a permutation: . _6P_6 \:=\:720 ways.


    Can you see why the answers are the same?

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