Stars and bars

A solution corresponds to a row of $10$ stars and $2$ bars: the stars before the first bar count $x$, those between the bars count $y$, those after the second bar count $z$. A row has $12$ positions, and we choose which $2$ are bars. The number of non-negative solutions is
$$\binom{12}{2}=66.$$

For positive solutions, set $x'=x-1$, $y'=y-1$, $z'=z-1\ge0$. Then $x'+y'+z'=7$, which has $\binom{9}{2}=36$ non-negative solutions.