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:

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||ooo
|o|oo

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, 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 $k-1$ bars among $n+k-1$ slots, since we need only $k-1$ bars to partition the space into $k$ summands. Therefore the solution is $N={n+k-1 \choose k-1}$, if $N$ denotes the number of unique sums.

For the next part, set aside $k$ 'o's since each partition has to have at least $1$ 'o'. Now the problem reduces to the previous one with a reduced value of $n$ i.e. $n-k$. So the solution is

$N={n-k+k-1 \choose k-1}={n-1 \choose k-1}$,

if $N$ denotes the number of unique sums.