Question:
There are 18 points in a plane of which 5 are collinear. How many straight lines can be formed by joining them?
Solution:
A line is formed by joining two points.
If the total number of points is 18 , the total number of lines would be $={ }^{18} \mathrm{C}_{2}$
But 5 points are collinear, so the lines made by these points are the same and would be only $1 .$
Hence there is 1 common line joining the 5 collinear points.
As these 5 points are also included in 18 points so these must be subtracted from the total case, i.e. ${ }^{5} \mathrm{C}_{2}$ must be subtracted from ${ }^{18} \mathrm{C}_{2}$.
Finally, the number of straight line $={ }^{18} \mathrm{C}_{2}-{ }^{5} \mathrm{C}_{2}+1$
= 144 lines