Question

There are 10 points in a plane out of which 4 points are collinear. How many straight lines can be drawn by joining any two of them?

• A. 39
• B. 40
• C. 45
• D. 21

Hint:

When dealing with collinear points, subtract the extra lines that would have been counted as distinct, as they lie on the same line.

Answer 1

The Correct Option is B

To determine the number of straight lines that can be drawn from 10 points where 4 are collinear, we can use the following approach:

• First, calculate the total number of straight lines that can be formed by selecting any 2 points from the 10 points. This is computed using the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of points, and \( r \) is the number of points to choose.
  Here, n = 10 and r = 2.

So, the total number of straight lines: \[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]

• Next, remove the lines that would be counted multiple times from the collinear points. Since there are 4 collinear points, the number of lines that can be drawn among them, which overlap as they are collinear, is: \[ \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \]

• Only 1 line can be drawn through collinear points, so remove (6 - 1 = 5) lines.

Thus, the number of distinct lines that can be drawn is: \[ 45 - 5 = 40 \]

In conclusion, 40 straight lines can be drawn by joining any two of the 10 points in the plane.

Answer 2

The total number of straight lines that can be formed by joining any two points from 10 points is given by the combination formula: \[ \text{Total lines} = \binom{10}{2} = \frac{10(10-1)}{2} = 45 \] However, 4 points are collinear, meaning they lie on the same straight line, and we should subtract the number of lines formed by these 4 points. The number of lines formed by the 4 collinear points is: \[ \text{Lines formed by collinear points} = \binom{4}{2} = \frac{4(4-1)}{2} = 6 \] Since all 4 collinear points form only 1 straight line, we subtract the extra 5 lines that would be counted if we treated them as distinct. Thus, the total number of distinct lines is: \[ \text{Total distinct lines} = 45 - 5 = 40 \] Therefore, the number of straight lines that can be drawn is 40.