Question

Consider a rectangle ABCD having $5,7,6,9$points in the interior of the line segments$AB,CD,BC,andDA$ 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

$717$

The explanation for the correct option:

Step 1. Given data of a rectangle :

Given that a rectangle $ABCD$has $5,7,6,9$points in the interior of the line segments $AB,CD,BC,andDA$ respectively

$\alpha$ be the number of triangles having these points as vertices.

$\begin{array}{rcl}\alpha & =& {}^{6}C_1\times {}^{7}C_1\times {}^{9}C_1+{}^{5}C_1\times {}^{7}C_1\times {}^{9}C_1+{}^{5}C_1\times {}^{6}C_1\times {}^{9}C_1+{}^{5}C_1\times {}^{6}C_1\times {}^{7}C_1\\ & =& 378+315+270+210\\ & =& 1173\end{array}$

$\beta$be the number of quadrilaterals having these points from different sides as vertices.

$\begin{array}{rcl}\beta & =& {}^{5}C_1\times {}^{6}C_1\times {}^{7}C_1\times {}^{9}C_1\\ & =& 5\times 6\times 7\times 9\\ & =& 1890\end{array}$

Step2. Calculate $(\beta -\alpha )$ :

$\begin{array}{rcl}(\beta -\alpha )& =& 1890-1173\\ & =& 717\end{array}$

Hence, the correct option is (C).