Question

Consider a rectangle $ABCD$ having $5,7,6,9$ points in the interior 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 :

• A. $1890$
• B. $795$
• C. $717$
• D. $1173$

Answer 1

The correct option is C $717$

$\alpha ={}^6{C}_1\cdot {}^7{C}_1\cdot {}^9{C}_1+{}^5{C}_1\cdot {}^7{C}_1\cdot {}^9{C}_1+{}^5{C}_1\cdot {}^6{C}_1\cdot {}^9{C}_1+{}^5{C}_1\cdot {}^6{C}_1\cdot {}^7{C}_1$
$=378+315+270+210=1173$

$\beta ={}^5{C}_1\cdot {}^6{C}_1\cdot {}^7{C}_1\cdot {}^9{C}_1=1890$
$\Rightarrow \beta -\alpha =1890-1173=717$