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Math Help - in how many arrangements can be made

  1. #1
    Super Member bigwave's Avatar
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    Cool in how many arrangements can be made

    In how many arrangements can 7 men and 7women be placed in a line if the women and men alternate.

    I thought the would 14! divided by something but not finding it...
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    \boxed{MW}\;\boxed{MW}\;\ldots \;\boxed{MW}\rightarrow{7!\cdot}7!

    \boxed{WM}\;\boxed{WM}\;\ldots \;\boxed{WM}\rightarrow{7!\cdot}7!

    ...

    Fernando Revilla
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  3. #3
    Super Member bigwave's Avatar
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    (7!7!)2 = 50,503

    so you mean that (7!7!)2 = 50,503

    btw how did you put a box accent around your text?
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  4. #4
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    He used "\boxed" in LaTex. You can see the LaTex code for any expression by double clicking on it.

    One way to get that answer is to think of putting the 7 men in a row- there are 7! ways to do that- then put the 7 women in a row- there are 7! ways to do that. Then place the women, in that order, between each of the men in the row. Since any of the 7! ways of ordering the men can be matched with any of the 7! ways of ordering the women, there are (7!)(7!) ways to do that.

    Another way to see that is to first pair each man with a woman- because of the "MW" boxes I suspect this is what FernandoRevilla did. There are 7 ways to pair a woman with the first man, 6 ways with the second, etc. so 7! pairs. There are then 7! ways to order those pairs.

    However, that is always with a man first. If, instead, we put a woman first, then a man, etc., we have a second way for each such combination. I think the correct answer is (2)(7!)(7!).

    I realize now that I misread the previous posts and that (2)(7!)(7!) was given all along.
    Last edited by HallsofIvy; December 27th 2010 at 03:29 AM.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by bigwave View Post
    so you mean that (7!7!)2 = 50,503
    Right, (7!7!)2 =\ldots=?

    btw how did you put a box accent around your text?
    For example, if you write \boxed{\int_a^bf(x)\;dx} it will appear:

    \boxed{\int_a^bf(x)\;dx}

    Fernando Revilla

    Edited: Sorry, I didn't see the previous post.
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