Consider a rectangle $$ABCD$$ having 5, 6, 7, 9 points in the interior of the line segments $$AB$$, $$BC$$, $$CD$$, $$DA$$ respectively. Let $$\alpha$$ be the number of triangles having these points from different sides as vertices and $$\beta$$ be the number of quadrilaterals having these points from different sides as vertices. Then $$(\beta - \alpha)$$ is equal to:

We have a rectangle $$ABCD$$ with 5, 6, 7, 9 points in the interior of sides $$AB$$, $$BC$$, $$CD$$, $$DA$$ respectively. We need to find $$\beta - \alpha$$, where $$\alpha$$ is the number of triangles and $$\beta$$ is the number of quadrilaterals formed by choosing vertices from different sides.

For a triangle, we choose 3 sides out of 4 and then one point from each chosen side. The number of such triangles is:

$$\alpha = 5 \cdot 6 \cdot 7 + 5 \cdot 6 \cdot 9 + 5 \cdot 7 \cdot 9 + 6 \cdot 7 \cdot 9 = 210 + 270 + 315 + 378 = 1173$$

For a quadrilateral, we choose one point from each of the 4 sides:

$$\beta = 5 \cdot 6 \cdot 7 \cdot 9 = 1890$$

Therefore, $$\beta - \alpha = 1890 - 1173 = 717$$.