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

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

 

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