How many different straight lines can be formed by joining 12 different points on a plane of which 4 art collinear and the rest are non-collinear?

Options

A. 16
B. 32
C. 61
D. 64

Answer

To solve the problem of how many different straight lines can be formed by joining 12 different points on a plane, where 4 points are collinear and the remaining 8 points are non-collinear, we can break it down into three cases.

### Step-by-Step Solution:

1. **Identify the Points**:
- We have a total of 12 points.
- Out of these, 4 points are collinear (let's denote them as A, B, C, D).
- The remaining 8 points are non-collinear (let's denote them as P1, P2, P3, P4, P5, P6, P7, P8).

2. **Case 1: Lines from Non-Collinear Points**:
- To form a straight line from the 8 non-collinear points, we can choose any 2 points.
- The number of ways to choose 2 points from 8 is given by the combination formula:
\[ \text{Number of lines} = \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

3. **Case 2: Lines from Collinear Points**:
- Since all 4 collinear points lie on the same line, they can only form 1 unique line regardless of how many points we choose from them.
- Thus, the number of lines formed from collinear points is:
\[ 1 \]

4. **Case 3: Lines from One Collinear Point and One Non-Collinear Point**:
- We can also form lines by selecting one point from the collinear points (4 choices) and one point from the non-collinear points (8 choices).
- The number of lines formed in this case is:
\[ 4 \times 8 = 32 \]

5. **Total Lines**:
- Now, we sum up the lines from all three cases:
\[ \text{Total lines} = \text{Lines from non-collinear points} + \text{Lines from collinear points} + \text{Lines from one collinear and one non-collinear point} \]
\[ \text{Total lines} = 28 + 1 + 32 = 61 \]

### Final Answer:
The total number of different straight lines that can be formed is **61**.

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