Puzzles/Statistical puzzles/Summing n/Solution

The solution is easily found if the problem is restated graphically. Consider n=3, k=2. Possible sums are 3 = 0+3, 3 = 1 + 2. A graphical representation could be:


||ooo
|o|oo

, respectively, with the bars representing separations between the summands and 'o's representing the value of the summands. Then obviously the problems is equivalent to finding the number of ways to distribute bars among slots, since we need only bars to partition the space into summands. Therefore the solution is , if denotes the number of unique sums.

For the next part, set aside 'o's since each partition has to have at least 'o'. Now the problem reduces to the previous one with a reduced value of i.e. . So the solution is

,

if denotes the number of unique sums.