Consider a rectangle ABCD having 5, 7, 6, 9 points in the integer of the line segments AB, CD, BC, 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:

 No of points on AB = 5

                             BC = 6

                             CD = 7

                             AD = 9

Number if triangles with vertices on different sides   

$$\begin{gathered} =(5\times 6\times 7)+(6\times 7\times 9)+(5\times 7\times 9)+(5\times 6\times 9) \\ =210+376+315+270=1173 \end{gathered}$$

B = Number of  quadrilaterals whose vertices lie on different sides

B = $5\times 6\times 7\times 9$

B = 1890

$\beta -\alpha$ = 1890-1173=717