Question

If there are $12$ points in a plane out of which only $5$ are collinear, then the number of quadrilaterals that can be formed using these points is

• A. $210$
• B. $280$
• C. $350$
• D. $420$

Answer 1

The correct option is D $420$

Given : $12$ points in a plane out of which only $5$ are collinear.
Number of non-collinear points $=7$
Number of quadrilaterals
$=$ (select $4$ points from $7$ points) $+$ (select $3$ points from $7$ points and select $1$ point from $5$ collinear points)
$+$ (select $2$ points from $7$ points and select $2$ points from $5$ collinear points)

Number of quadrilaterals
$={}^7{C}_4+{}^7{C}_3\times {}^5{C}_1+{}^7{C}_2\times {}^5{C}_2={}^7{C}_4[1+5]+\frac{7\times 6\times 10}{2}=\frac{7\times 6\times 5\times 6}{6}+210=210+210=420$