What is the number of triangles that can be formed be choosing the vertices from a set of 12 points in a plane,seven of which lie on the same striaght line?

Options

A. 185
B. 175
C. 115
D. 105

Answer

To solve the problem of finding the number of triangles that can be formed by choosing vertices from a set of 12 points in a plane, where 7 of the points lie on the same straight line, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Points and Collinear Points**:
- We have a total of 12 points.
- Out of these, 7 points are collinear (lie on the same straight line).
- The remaining points (12 - 7 = 5) are non-collinear.

2. **Calculate the Total Ways to Choose 3 Points**:
- To form a triangle, we need to select 3 points from the total of 12 points.
- The number of ways to choose 3 points from 12 is given by the combination formula \( C(n, r) \), which is calculated as:
\[ C(12, 3) = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]

3. **Calculate the Ways to Choose 3 Collinear Points**:
- Since 7 points are collinear, we can also choose 3 points from these collinear points.
- The number of ways to choose 3 points from 7 is:
\[ C(7, 3) = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

4. **Subtract the Collinear Combinations from the Total**:
- To find the number of triangles that can be formed using non-collinear points, we subtract the number of ways to select 3 collinear points from the total ways to select 3 points:
\[ \text{Number of triangles} = C(12, 3) - C(7, 3) = 220 - 35 = 185 \]

5. **Conclusion**:
- Therefore, the number of triangles that can be formed by choosing the vertices from the given set of points is **185**.