Question

The number of triangles that can be formed by choosing the vertices from a set of $12$ points, seven of which lie on the same straight line, is

• A. $185$
• B. $175$
• C. $115$
• D. $105$

Answer 1

The correct option is A $185$

Triangles can be formed by joining three non-collinear points.

We can form triangles by choosing any $3$ points from given $12$ points.

But as there are $7$ points which lying on the straight line, we will get the no.of triangles formed by subtracting the number of triangles formed by joining these collinear points.

Total number of ways of selecting any $3$ points is ${}^{12}{C}_3$.

Therefore, required number of triangles $={{}^{12}C}_3-{{}^{7}C}_3=220-35=185$.