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Fixed sum of combinations

  1. Nov 28, 2013 #1
    When you have combinations where digits are 0,1,2...,m, meaning we have n=m+1 and k, is there a way to see how much of them sum up to a given number? For the sake of simplicity I have the digits 0,1,2...,7 (so n=8), and k=3. I need to find how much of these combinations WITH repetition sum up to 7. Is there a formula for this?
     
  2. jcsd
  3. Nov 28, 2013 #2
    By sum up, I mean the sum of all 3 digits in each combination needs to be equal to 7.
     
  4. Nov 29, 2013 #3

    Stephen Tashi

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    I don't know a formula, but there is a procedure - or at least a concise way to phrase the problem.

    Compute

    [itex] (1 + x + x^2 + x^3 +x^4 + x^5 + x^6 + x^7)^3 = ? [/itex]

    and then look at the coefficient of [itex] x^7 [/itex] in the answer. The coefficient counts the number of permutations of the numbers 0,1,2...7 that add to 7. To get combinations, divide that by [itex] 3! [/itex].
     
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