3) I am not sure I understood this question. Please explain.
Should have been obvious.
3a) How many ways can we get distinct non-negative integers (x1,x2,x3,x4...xr) such that x1+x2+x3..xr = N where N is another distinct non-negative number.
Solved the same way as separating N identical balls into R urns.
b) Here we just add the condition that x1,x2,x3 are all greater than 0.
We can again solve this as the urns+balls problem i.e the number of ways of placing N balls into R urns with each urn having at least one ball. Keep the N balls in a row. We get N-1 spots between the balls which is a demarcator. We'll now choose (R-1) spots from the N-1 spots.
Answer is (n-1)C(r-1) i.e (n-1)!/(n-r)!(r-1)!