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Math Help - number of ways ?

  1. #1
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    number of ways ?

    In how many ways can we place n+x balls in n boxes
    with a condition that at least 1 ball is present in every box
    when
    1)balls r identical
    2)balls r numbered from 1 to n+x
    and x<n
    Explain ur answer
    Last edited by mr fantastic; December 14th 2011 at 07:19 PM. Reason: Title.
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  2. #2
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    Re: number of ways ????

    Quote Originally Posted by livinggourmand View Post
    In how many ways can we place n+x balls in n boxes
    with a condition that at least 1 ball is present in every box
    when
    1)balls r identical
    2)balls r numbered from 1 to n+x
    and x<n
    Explain ur answer
    You failed to tell us if the boxes are all different.
    So we assume they are.

    There are \binom{K+N-1}{K} ways to place K identical items into N different cell.
    If we require that no cell is empty then it must be that case that K\ge N and that multi-selection formula becomes
    \binom{K-1}{K-N}.

    For part b), we need to count the number of surjections (onto functions) from a set of N+x to a set of N.

    If K\ge N and then the number of surjections from a set of K to a set of N is \sum\limits_{j = 0}^N {( - 1)^j \binom{N}{j} (N - j)^K } .
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  3. #3
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    Re: number of ways ????

    n if boxes are identical??
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  4. #4
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    Re: number of ways ????

    Quote Originally Posted by livinggourmand View Post
    n if boxes are identical??
    If the boxes are identical then it becomes more difficult.

    For part a), we have to count the numbers of ways to have K summands for the integer N+x. That is not easy.

    Part b) is a bit easier that a). We count the number of unordered partitions of N+n individuals into N groupings.

    However, I suspect that whoever wrote this question meant the boxes all different. Because, that is an easier question.
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