Question

There are n points in a plane of which p points are collinear. How many lines can be formed from these points?
A. $^n{C_2}{ - ^P}{C_2}$
B. $^n{C_2}{ - ^p}{C_2} + 1$
C. $^n{C_2}{ - ^p}{C_2} - 1$
D.$^{n - p}{C_2}$

Answer 1

Hint-Make use of the formula that if there are n points, then the number of lines would be $^n{C_2}$

Given that there are n points ,
So, from this the number of lines that can be formed is $^n{C_2}$
But, given that P points are collinear , so from this if P points are collinear , the number of lines formed from this collinear points would be $^p{C_2}$
But since the points are collinear, they from only one line
So, the total number of lines that can be formed will be the difference between the number of lines that can be formed from n points and the number of lines that can be formed from the P points that are collinear plus the one line which is formed from the collinear points
So, if we represent this in the form of equation , we can write this as
$^n{C_2}$-$^p{C_2}$+1
So, if we look at the options, option B is the correct answer.

Note: We have added 1 in the equation because the collinear points would definitely form one straight line ,because even after we find the difference we would still have one line remaining