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 P: 2,490 If I understand you, you want to know the probability of getting a sum = n for in k tries where n is a positive integer and the regions are the natural numbers: 1, 2, 3, ..........n. This involves compositions of integers which you demonstrated. The formula for the number of compositions for any natural number n is $2 ^ {n-1}$. Since you are only interested in the compositions of three integers (k=3) and are not including zero, the formula is (n-1)!/(k-1)!(n-k)! How would you use this information to find the probability for n=9, k=3?