$\binom{n-1}{r-1}$ distinct positive integer-valued vectors $(x_1,x_2, ... x_r)$...

$\binom{n-1}{r-1}$ distinct positive integer-valued vectors $(x_1,x_2, ...
x_r)$...

Proposition: There are $\binom{n-1}{r-1}$ distinct positive integer-valued
vectors $(x_1,x_2, ... x_r)$ satisfying the equation
$$x_1+x_2 + ... x_r = n \ \ \ \ \ \ \ \ \ \ x_i\gt0$$
Textbook: To obtain the number of nonnegative (as opposed to positive)
solutions, note that the number of nonnegative solutions of $x_1+x_2 + ...
x_r = n$ is the same as the number of positive solutions of $y_1 + ... +
y_r = n+r$, as seen by letting $y_i = x_i +1,i = 1,...,r$.
Question: I see that $y_1 + ... + y_r = x_1+x_2 + ... x_r +
\sum_{i=1}^{r}1 = n+r$, but why must the number of nonnegative solutions
of $x_1+x_2 + ... x_r = n$ be the same as the number of positive solutions
of $y_1 + ... + y_r = n+r$? And what exactly is a nonnegative vector
solution, is it a vector such that no scalar is less than zero?