Question

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

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

Answer 1

Answer: (A)

$185$

Total no. of points $=12$

No. of points in same line $=7$

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

${}^n{C}_r=\frac{n!}{r!(n-r)!}$

No. of triangles $=\frac{12!}{3!9!}-\frac{7!}{3!4!}=\frac{12\times 11\times 10}{3\times 2}-\frac{7\times 6\times 5}{3\times 2}$

$=220-35$

$=185$ total triangles.