Let me motivate my general question with an explicit example:

Suppose I am looking for all unique combinations of exactly three non-negative integers that sum to five. The solutions are 005, 014, 023, 113, and 122. Which means that there are five unique combinations.

Is there a way to find the $\textit{number}$ of unique combinations of exactly $k$ non-negative integers that sum to $n$? I'd rather not generate all the unique combinations and then count. I am hoping that there is a straightforward combinatoric solution to this.

Please let me know if more clarification is needed.

Thanks!