Question

The maximum number of triangles that can be formed by choosing the vertices from a set of $12$ points, $7$ of which are in a straight line, are

• A. $84$
• B. $175$
• C. $185$
• D. $105$

Answer 1

The correct option is C $185$

Total no. of points$=12$

No. of points in same line$=7$

No. of triangles formed${=}^{12}{C}_3{-}^7{C}_3$

${}^{12}{C}_3=\frac{12!}{3!9!}=220$

${}^7{C}_3=\frac{7!}{3!4!}=35$

So, no. of triangles formed$=220-35=185$