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Combinatorics Path question

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Let a, b, and c be positive integers. How many paths are there from (0, 0, 0) to
    (a, b, c) if we are only allowed to increase one of the coordinates by one at each
    step?


    2. Relevant equations



    3. The attempt at a solution

    This problem is easy for the path between (0,0) to (a,b)
    because you can make the path into a binomial sequence, where if you increase a, then that will be a one, and if you increase b, then it will be a zero.

    so for two:
    total number of digits in the binary sequence= a+b
    where a= # of 1s and b=#of 0's

    So the total # of paths=
    (a+b) choose a, which is equivalent to (a+b) choose b.

    However for (0,0) to (a,b,c)
    the total digits in the binary= a+b+c
    and define the number of ones as a
    and non-ones (zeros)= a+b

    which yields:
    Total path #s= (a+b+c) choose a.

    I think the answer is:

    ((a+b+c) choose a)((b+c) choose b)

    However, how do I explain that? Can i define c as the number of two's in the binary sequence? I thought binaries only had ones and zeros, that is why I haven't done that. Any help would be great!
     
  2. jcsd
  3. Sep 22, 2009 #2
    This can be done using a multinomial formula.

    But yes

    [tex]\left( \begin{array}{c}{a+b+c}&{a}\end{array} \right) \left( \begin{array}{c}{b+c}&{b}\end{array} \right)[/tex]

    is fine.

    --Elucidus
     
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