What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.) $\textbf{(A) }2\qquad$ $ \textbf{(B) }3\qquad$ $ \textbf{(C) }4\qquad$ $ \textbf{(D) }5\qquad$ $ \textbf{(E) }6\qquad$