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Math Help - Ordered Partitions and Distributions

  1. #1
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    Ordered Partitions and Distributions

    Determine the number of integer solutions of X1 +X2 +X3 +X4 +X5 < 40
    where

    a) Xi > 0, 1 <= i <= 5
    b) Xi >= -3, 1 <= i <= 4 and X5 => 3
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  2. #2
    Moo
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    Hello,

    The method I'll present may look a tad weird, and experimental, so don't hesitate if you have any objection lol.

    You basically need the two theorems in there : Stars and bars (probability) - Wikipedia, the free encyclopedia

    First let X_6 a positive integer (>0) and Y_6 a nonnegative integer (\geq 0)

    For question 1), write the inequality as X_1+X_2+X_3+X_4+X_5+X_6=40 (think on your own to know why it's correct) and apply theorem 1 (because we have strict inequalities for the unknowns).

    For question 2), we have X_1+X_2+X_3+X_4+X_5+Y_6=39

    Let Y_i=X_i+3 ~,~ i\in\{1,2,3,4\} and Y_5=X_5-3

    Substitute the X_i by the Y_i in the inequality to get :
    Y_1+Y_2+Y_3+Y_4+Y_5+Y_6=39, where Y_i\geq 0 ~,~ i\in\{1,2,3,4,5,6\}

    and you can apply theorem 2.
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