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 [itex] 2 ^ {n1} [/itex]. Since you are only interested in the compositions of three integers (k=3) and are not including zero, the formula is (n1)!/(k1)!(nk)!
How would you use this information to find the probability for n=9, k=3?
