Question

Out of 18 point in a plane no three are in the same straight line except five points which are collinear. how many triangles can be formed by joining them ?

Answer 1

Given that,

Total points $=18$

collinear points $=5$

Number of lines formed

Given that,

Total point $=18$

Collinear points $=5$

Number of triangles formed from $18$ points ${=}^{18}{C}_3$

( Since we require three points to form a triangle )

Five points are collinear ${=}^5{C}_3$

$\Rightarrow$ Required number of triangles ${=}^{18}{C}_3{-}^5{C}_3+1$

${}^{18}{C}_3=\frac{18!}{15!\times 3!}=\frac{18\times 17\times 16\times 15!}{15!\times 3\times 2}=816$

${}^5{C}_3=\frac{5!}{2!\times 3!}=\frac{5\times 4\times 3!}{2!\times 3!}=10$

${\Rightarrow }^{18}{C}_3{-}^5{C}_3$

$\Rightarrow 816-10$

$\Rightarrow 806$