There are 12 points in a plane in which 6 are collinear. Number of different straight lines that can be drawn by joining them, is

Options

A. 51
B. 52
C. 132
D. 18

Answer

To solve the problem of finding the number of different straight lines that can be drawn by joining 12 points in a plane, where 6 of these points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Total Points**:
We have a total of 12 points in the plane.

2. **Choosing Points to Form Lines**:
A straight line can be formed by joining any two points. The number of ways to choose 2 points from 12 can be calculated using the combination formula:
\[ \text{Total lines} = \binom{12}{2} = \frac{12 \times 11}{2} = 66 \]

3. **Accounting for Collinear Points**:
Among the 12 points, 6 points are collinear. This means that if we choose any 2 points from these 6, they will not form distinct lines since they all lie on the same line. The number of ways to choose 2 points from these 6 collinear points is:
\[ \text{Collinear lines} = \binom{6}{2} = \frac{6 \times 5}{2} = 15 \]

4. **Adjusting for the Collinear Points**:
Since all 6 collinear points form only one line, we need to subtract the number of lines formed by choosing 2 points from these 6 from our total. However, we need to add back 1 because all 6 points together form a single line. Thus, the adjustment is:
\[ \text{Distinct lines} = \text{Total lines} - \text{Collinear lines} + 1 \]
Plugging in the values:
\[ \text{Distinct lines} = 66 - 15 + 1 = 52 \]

5. **Final Answer**:
Therefore, the total number of different straight lines that can be drawn by joining the 12 points is:
\[ \boxed{52} \]