Question

i understand how to count the solutions in nonnegative integers x1,.....xk to x1+.....+xk = n but how about x1,....xk to x1+....+xk =< n?

Answer 1

mathematics induction? shocks. i'm weak of it. . lol

Answer 2

i assume induction is possible, but my guess is that since x1+...xk is less than or equal to n, if we subtract the difference between those, then maybe, instead of (n+k-1)C(k-1), i can say (n-r+k-1)C(k-1), where r is n-(x1+....+xk), but thats only my guess.

Answer 3

perhaps, because one way of diminishing the value is through subtraction. .