jmc

The sides of the triangle must satisfy the triangle inequality, so $AB+AC>BC$, $AB+BC>AC$, and $AC+BC>AB$. Substituting the side lengths, these inequalities turn into $$\begin{array}{rl}(3n-3)+(2n+7)& >2n+12,\\ (3n-3)+(2n+12)& >2n+7,\\ (2n+7)+(2n+12)& >3n-3,\end{array}$$ which give us $n>8/3$, $n>-2/3$, and $n>-22$, respectively.

However, we also want $\angle A>\angle B>\angle C$, which means that $BC>AC$ and $AC>AB$. These inequalities turn into $2n+12>2n+7$ (which is always satisfied), and $2n+7>3n-3$, which gives us $n<10$.

Hence, $n$ must satisfy $n>8/3$ and $n<10$, which means $$3\le n\le 9.$$ The number of positive integers in this interval is $9-3+1=\boxed{7}$.