Question

There are 12 points in a plane of which 5 are collinear. These points are joined in pairs. Find the number of straight lines formed.

Answer 1

Hint: - To solve this question first we have to know if we have given n points and we have to make a straight line joining any two of them then the number of ways will be $^n{C_2}$. And we should also know that on joining collinear points we get one and only one line.

Complete step-by-step answer:
We have been given 12 points in a plane and in which 5 are collinear.
As we understand in hint if we have given n points and we have to make a straight line joining any two of them then the number of ways will be $^n{C_2}$.
That means if we have 12 points and none of them is collinear then on joining any two of them the number of straight lines will be $^{12}{C_2}$.
And we also know collinear points means points lying in the same line that means on joining collinear points we will get one and only one line and hence,
We have given 5 collinear points so on joining these 5 points we will get only one line.
And hence for the total number of straight lines we have to subtract $^5{C_2}$and add 1 in place of $^5{C_2}$ because only one line will form from these 5 points.
Total number of straight lines formed will be
$^{12}{C_2}{ - ^5}{C_2} + 1$= 57.
Here $^{12}{C_2}$ is for number of straight lines formed if no any collinear point is present but we subtract $^5{C_2}$ because there are 5 collinear points given and we add 1 because on joining collinear points only one line will be formed.

Note: -Whenever we get this type of question the key concept of solving is we have to understand laws of permutation and combination and also understand number of lines formation from general points and collinear points then we will be able to answer this type of questions.