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Math Help - Combinations with constraint

  1. #1
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    Combinations with constraint

    Dear all,

    I have a problem of combinatorial/permutation in the context of computer networks. To easy of explanation, suppose that you have vector with n=3 elements, say [1,3,4].
    The summation of the n elements yields a total T = 8.

    Now consider these constraints:
    a) Each element in the vector may vary from 0 to 4.
    b) The summation of the n elements MUST be equal to the total T = 8.

    The question is, how many combination do we can do with that vector, considering the constraints?

    Valid results include:

    1, 3, 4
    1, 4, 3
    3, 1, 4
    3, 4, 1
    4, 1, 3
    4, 3, 1
    2, 2, 4
    2, 3, 3
    0, 4, 4


    Many thanks in advance for your help
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  2. #2
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    There are just four selections of three having the sum eight.
    [0,4,4],~[2,2,4]~ ,~[2,3,3]~\&~[1,3,4]
    The first three can be arranged is three ways, while the last has six rearrangements.

    Or we can expand the expression \left( {\sum\limits_{k = 0}^4 {x^k } } \right)^3 , the coefficient of x^8 is 15 that is the same answer.
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  3. #3
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    Thank you for your reply. However, I still didn't get it!

    In the real problem, I'll have vectors of up to 5k elements.

    How did you get the number of selections having the sum eight?

    Thanks.
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