Question

Find the number of straight lines obtained by joining $10$ points on a plane, if
(i) No three of which are collinear
(ii) Four points are collinear

Answer 1

(i) Number of straight line by joining $10$ pt. S.T. to three of which are collinear is same as choosing $2$ pt. Out of $10$.

So, no. of ways$={}^{10}{C}_2=\frac{10!}{2!8!}=\frac{10\times 9\times 8!}{2}=45$

(ii) When $4$ Pts. are collinear then total no. of lines$={}^{10}{C}_2$ but it also contains lines using that $4$ collinear pt. So, we need to substract that line and add $1$ as collinear pts make only one line.

Lines using $4$ collinear pt $={}^4{C}_2$

No. of ways is $4$ pts are collinear$={}^{10}{C}_2-{}^4{C}_2+1$

$=45-\frac{4!}{2!2!}+1=46-6=40$.