Question

There are 16 points in a plane out of which 6 are collinear, then how many lines can be drawn by joining these points.
A. $106$
B. $105$
C. $60$
D. $55$

Answer 1

Hint: The points that are collinear lie on the same straight line. The remaining points form lines with the collinear points. The remaining points also form lines among themselves. Using this information, we can find the total number of lines.

Complete step by step solution:
We are given that there are 16 points in a plane out of which 6 are collinear.
Collinear means that all 6 points lie on the same line. Therefore, we get only one straight line drawn from the 6 collinear points.
From each of the 6 points there will be another 10 lines joining the remaining 10 points. So, the number of lines is 6(10) =60
Now these 10 points will have lines among themselves. Since two points are required to make a line we have, ${}^{10}{C_2}$ lines.
Therefore, the total number of lines are $1 + 60 + {}^{10}{C_2}$
$1 + 60 + {}^{10}{C_2}$= $1 + 60 + \dfrac{{10!}}{{2!(10 - 2)!}}$
                        = $1 + 60 + \dfrac{{10!}}{{2!8!}}$
                        = $1 + 60 + \dfrac{{10 \times 9}}{2}$
                        = $1 + 60 + 45$
                        = $106$

Option ‘A’ is correct

Note: To solve this problem one must note that collinear points are the points which lie on the same line and two points are required to form a line.
If there are n points in the plane then the number of lines formed is given by ${}^n{C_2}$.
If there are a total of n points in the plane out of which m are collinear then we can say that number of lines formed is given by the formula ${}^n{C_2} - {}^m{C_2} + 1$ as we need two points to form a line.