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Math Help - Divide N balls to subgroups

  1. #1
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    Divide N balls to subgroups

    Hello.
    I am trying to prove that the number of options to divide N balls to at most k groups equals the number of options to divide N balls to groups in a way that each group is with at most k balls.

    The balls are indistinguishable, no empty groups are allowed and the order of the groups does not matter.

    Any idea?

    Thanks!
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  2. #2
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    Quote Originally Posted by guyov1 View Post
    Hello.
    I am trying to prove that the number of options to divide N balls to at most k groups equals the number of options to divide N balls to groups in a way that each group is with at most k balls. The balls are indistinguishable, no empty groups are allowed and the order of the groups does not matter.
    I have read and reread your question several times.
    Each time I get a different way of seeing it.
    So I am not really sure what it is asking.

    I think you can use Stirling numbers of either the first or second kind.
    Stirling Number of the Second Kind -- from Wolfram MathWorld
    Stirling Number of the First Kind -- from Wolfram MathWorld

    Once you look into that, perhaps you could clarify your question.
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  3. #3
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    Quote Originally Posted by guyov1 View Post
    Hello.
    I am trying to prove that the number of options to divide N balls to at most k groups equals the number of options to divide N balls to groups in a way that each group is with at most k balls.

    The balls are indistinguishable, no empty groups are allowed and the order of the groups does not matter.

    Any idea?

    Thanks!
    Hi guyov1,

    OK, letís take a definite example. Suppose you have 20 balls and you want to divide them into no more than 5 groups. For example, one way to do the division would be to have groups of size 10, 5, 3, 1, 1. (Iím deliberately listing the groups in order of decreasing size, which is OK because the order of the groups does not matter.) We can visualize the arrangement like this:

    Code:
      XXXXXXXXXX
      XXXXX
      XXX
      X
      X
    Now rotate the figure.
    Code:
      XXXXX
      XXX
      XXX
      XX
      XX
      X
      X
      X
      X
      X
    This figure corresponds to a division into 10 groups, of size 5, 3, 3, 2, 2, 1, 1, 1, 1, 1.

    The arrangement in the first figure will have no more than 5 groups if and only if the arrangement in the rotated figure has no groups of size larger than 5.

    Get the idea?
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