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

$185$

Explanation for the correct option:

Calculate the number of triangles that can be formed.

We are given, $12$ set of points.

Number of triangles formed with $12$ points $={}^{12}\mathrm{C}_3$

But, according to the given condition, $7$ points lie on the same straight line

Thus, the selection of $3$ points out of $7$ collinear points $={}^7\mathrm{C}_3$, which we need to deduct from the non-collinear points.

Thus, the required number of triangles $={}^{12}\mathrm{C}_3–{}^7\mathrm{C}_3=220-35=185$

Hence, Option $\left(\mathrm{A}\right)$ is the correct answer.