(Feller Introduction to Probability Theory and Its Applications - Volume I - 3rd edt.)
In his book, Feller presents the following formula for taking a sample of size $r$ without replacement from a population of size $n$ so that $N$ given (already chosen) elements will be in the sample: $$\binom{n - N}{r - N}.$$
For the with-replacement case, otherwise, he claims it can't be derived from a direct argument.
After, in the same charapter, he presents the following formula for sample with replacement: $$u(r,n) = \sum_{v = 0}^N (-1)^v \binom{N}{v} \left(1 - \frac{v}{n} \right)^r.$$
My question is:
Why can't we just solve this problem as the number of natural solutions of: $$x_1 + ... + x_n = r$$ such that, WLOG, ${x_1, ..., x_N} \in \mathbb{N} \setminus \{0\}$ and ${x_{N+1}, ..., x_n} \in \mathbb{N}?$
Hence, the solution would be: $$\binom{n + r - N - 1}{r - N}.$$