There are 8 points in a plane out of which 4 are collinear. How many triangles can be formed with these these points as vertices?

Options

A. a. 52
B. b. 32
C. c. 35
D. d. 56

Answer

To find the number of triangles that can be formed with 8 points in a plane, out of which 4 are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a total of 8 points, and among these, 4 points are collinear. To form a triangle, we need to choose 3 points. However, if all 3 points are chosen from the 4 collinear points, they will not form a triangle.

2. **Case 1: Choosing 3 Points from Non-Collinear Points**:
- We have 4 non-collinear points. We can choose all 3 points from these non-collinear points.
- The number of ways to choose 3 points from 4 non-collinear points is given by the combination formula \( C(n, r) \), where \( n \) is the total number of points and \( r \) is the number of points to choose.
- Therefore, the number of triangles formed in this case is:
\[ C(4, 3) = \frac{4!}{3!(4-3)!} = 4 \]

3. **Case 2: Choosing 2 Points from Non-Collinear Points and 1 Point from Collinear Points**:
- We can choose 2 points from the 4 non-collinear points and 1 point from the 4 collinear points.
- The number of ways to choose 2 points from the 4 non-collinear points is \( C(4, 2) \), and the number of ways to choose 1 point from the 4 collinear points is \( C(4, 1) \).
- Therefore, the number of triangles formed in this case is:
\[ C(4, 2) \times C(4, 1) = \left(\frac{4!}{2!(4-2)!}\right) \times \left(\frac{4!}{1!(4-1)!}\right) = 6 \times 4 = 24 \]

4. **Total Number of Triangles**:
- Now, we add the number of triangles from both cases:
\[ \text{Total triangles} = \text{Triangles from Case 1} + \text{Triangles from Case 2} = 4 + 24 = 28 \]

### Final Answer:
The total number of triangles that can be formed is **28**.