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Math Help - combinatorial identity problem

  1. #1
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    combinatorial identity problem

    Suppose that n people are gathered for a game requiring two teams each with k players. assume 2k <= n. 1 team wear red shirts and the other team will wear blue shirts. the n-2k people not on either team will just watch. we can choose the teams in either of 2 ways. 1 choose k players to wear the red shirts then choose k others to wear blue shirts. 2 the other is to choose the 2k players and then choose k of them to be red and the other will be blue. prove via combinatorial identity that these 2 methods are the same.
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  2. #2
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    Quote Originally Posted by canyiah View Post
    Suppose that n people are gathered for a game requiring two teams each with k players. assume 2k <= n. 1 team wear red shirts and the other team will wear blue shirts. the n-2k people not on either team will just watch. we can choose the teams in either of 2 ways. 1 choose k players to wear the red shirts then choose k others to wear blue shirts. 2 the other is to choose the 2k players and then choose k of them to be red and the other will be blue. prove via combinatorial identity that these 2 methods are the same.
    The first method is \binom{n}{k}\binom{n-k}{k}.
    The second method is \binom{n}{2k}\binom{2k}{k}.
    Can you show they are the same?
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  3. #3
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    i dont see it probably by the algebra method it will look the same.
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  4. #4
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    Quote Originally Posted by canyiah View Post
    i dont see it probably by the algebra method it will look the same.
    I have no idea what that means.
    Can you put into standard language?
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  5. #5
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    ill try it later today thanks
    Last edited by canyiah; March 23rd 2010 at 12:01 AM.
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  6. #6
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    showed they were equal via algebraic method just substitute using combinatorial formula c(n,k) = n!/(k!(n-k)!)
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