jmc

Let $n$ be the length of the third side. Then by the triangle inequality, $$\begin{array}{rl}n+5& >7,\\ n+7& >5,\\ 5+7& >n,\end{array}$$ which tell us that $n>2$, $n>-2$, and $n<12$. Hence, the possible values of $n$ are 3, 4, 5, 6, 7, 8, 9, 10, and 11, for a total of $\boxed{9}$.