Question

Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (i) straight lines (ii) triangles can be formed by joining them ?

Answer 1

There are 18 points in a plane out of which 5 points are collinear

Then number of straight lines joining these points are

${\Rightarrow }^n{C}_2-{(}^p{C}_2-1)$

${\Rightarrow }^n{C}_2{-}^p{C}_2-1$ (where n = 18 p = 5)

${\Rightarrow }^{18}{C}_2{-}^5{C}_2+1$

$\Rightarrow \frac{18\times 17}{2}-\frac{5\times 4}{2}+1$

$\Rightarrow 144$

Number of triangle ${=}^{13}{C}_3$

$=\frac{13!}{3!10!}=\frac{13\times 12\times 11}{3\times 2}$

$=13\times 2\times 11$

$=13\times 22=806$