Question: Twenty lattice points are arranged along the edges of a $4 \times3$ rectangles as shown. How many triangles have all three of their vertices among these points?
I started off with getting the total number of points possible, which is simply $\binom {20}3$. To get the number of triangles formed, we can first consider the complementary case when the lines are collinear (i.e no triangles formed).
However, I'm having trouble evaluating how many collinear lines can be formed in a $4 \times 3$. I'm not sure how you would consider the tilted lines.
Yeah, I would do it the way you proposed.
Horiz: $4\binom{5}{3}$
Vert: $5\binom{4}{3}$
slope=$\pm 1: 2\big(2\binom{3}{3}+2\binom{4}{3}\big)$
slope=$\pm2,\pm\cfrac{1}{2}: 4\times 2$