Non-negative and positive. Isn't that pretty much the same thing?. Or do you just mean using 1 thrugh 9 instead of 0 through 9?.
Then you would have:
The coefficient of x^9 is 56, so there are 56 arrangements of the digits 1-9 that sum to 9.
You could use the generating function,
Then look at the coefficient of the x^9 term. Which is indeed 220.
Suppose you wanted to do w+x+y+z=9, but only use the numbers 1 through 5
The coefficient of x^9 is 52, so there are 52 ways to sum to 9 using the digits 1 through 5.