Class 11

Math

Algebra

Permutations and Combinations

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?

- $185$
- $175$
- $115$
- $105$

No. of points in same line $=7$

No. of triangles formed $=_{12}C_{3}−_{7}C_{3}$

$_{n}C_{r}=r!(n−r)!n! $

No. of triangles $=3!9!12! −3!4!7! =3×212×11×10 −3×27×6×5 $

$=220−35$

$=185$ total triangles.