How many distinct samples of size n are there when we sample from a population of size N with replacement?

Question

If the population is size N, and the sample size is n, how can I generalize this case:
I want to find the number of distinct samples of size n that can be selected from the population, WITH replacement.

Answer 1

To be found is the cardinality of the set of tuples $(n_1,\dots,n_N)$ where the $n_i$ are nonnegative integers that satisfy: $$n_1+\cdots+n_N=n$$

With stars and bars we find that this cardinality equals:$$\binom{n+N-1}{N-1}$$