I am aware the number of distinct non-negative integer-valued vectors $(x_1, x_2, ..., x_r)$ satisfying the equation $x_1 + x_2 + ... + x_r = n$ is given by
$$n+r-1 \choose r-1$$
However, is there a formula to compute the number of distinct combinations of non-negative integers $ \{x_1, x_2, ..., x_r\}$ that satisfy the equation $x_1 + x_2 + ... + x_r = n$?
In other words, for $n=r=2$, the former formula will return
$${2+2-1 \choose 2-1}={3 \choose 1}=3$$
conveying there are $3$ distinct non-negative integer-valued vectors that satisfy $x_1+x_2=2$, that is, $(1,1),(2,0)$ and $(0,2)$. However, I am seeking a formula that will convey there are $2$ distinct combinations of non-negative integers that satisfy $x_1+x_2=2$, that is, $\{1,1\}$ and $\{2,0\}$.
Does such a formula exist?
Thank you!