Two lines run from the top vertex of the triangle to its base, and another line runs from the bottom-left vertex to the middle of the right side. Find the total number of triangles.

With the top vertex: 4 lines leave it, and both the base and the slanting line cross all 4, so each gives $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$: 12. With the bottom-left vertex as a corner and the base and slanting line as two sides, the third side can be either inner line or the right side: 3. Total $12+3=15$.