Question

There are 15 points in a plane, no three of which are in a straight line, except 6, all of which are in a straight line. The number of straight lines which can be drawn by joining them is

• A.
• B.
• C.
• D.

Hint:

When points are collinear, subtract the number of lines formed by those points from the total number of lines.

Answer 1

The Correct Option is B

Step 1: Understand the number of straight lines.
The total number of straight lines that can be formed by 15 points (if no three points are collinear) is given by \( \binom{15}{2} \), which is the number of ways to select 2 points out of 15.
Step 2: Account for the collinear points.
6 of the points are collinear, meaning they lie on the same straight line.
The number of straight lines that can be formed by any two of these 6 points is \( \binom{6}{2} \).
Step 3: Subtract the overcounted lines.
Since the lines formed by these 6 points have been counted twice, we subtract \( \binom{6}{2} \) from the total number of lines. \[ \text{Total number of lines} = \binom{15}{2} - \binom{6}{2} \]
Step 4: Conclusion.
The correct number of straight lines is \( \binom{15}{2} - \binom{6}{2} \), which corresponds to option (b).