Question

Out of 18 points in a plane, no three are in the same line except 5 points which are collinear.Find the number of lines that can be formed joining the point.(from Permutations and Combinations)

Answer 1

First find all possible lines then subtract lines which lie on those five points...

To form a line..... There must be at joining of two points...

18c2 - 5c2

then add one line which passes through five collinear points...

Answer is 143+1=144

Answer 2

The total number of lines that can be drawn are 144.

• There are total 18 points in plane.
• 5 points from the 18 points are collinear.
• Now, to draw a line we select any two points in a plane from the given points.
• Total number of ways of drawing a line from 18 points $^{18}C_2$.
• Now, 5 points are collinear that means they will form the same line. SO we subtract the lines made by the collinear points.

Number of lines formed = $^5C_2$

• Therefore, total number of lines joined by the points

Number of lines = $^{18}C_2$ - $^5C_2$ + 1

(1 is added because all the collinear points will form one line)

Number of lines = 153 - 10 + 1 = 144

• Number of lines drawn from 18 points on plane of which 5 are collinear is 144.