The number of scalene triangles having all sides of integral lengths, and perimeter less than $13$ is: 
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 18$
We can write all possible triangles adding up to 12 or less
\[(2, 4, 5)=11\]
\[(3, 4, 5)=12\]
\[(2, 3, 4)=9\]
This leaves $\boxed{\textbf{(C)} \quad 3}$ scalene triangles.
-coolmath34
-rubslul
(If you see any cases I missed out, edit them in.)