The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight line is

Options

A. 105
B. 150
C. 175
D. 185

Answer

To solve the problem of finding the number of triangles that can be formed by choosing vertices from a set of 12 points, where 7 of those points are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a total of 12 points, out of which 7 points are collinear (lie on the same straight line). We need to find the number of triangles that can be formed using these points.

2. **Calculate Total Combinations of Points**:
To form a triangle, we need to select 3 points. If all points were non-collinear, the number of ways to choose 3 points from 12 is given by the combination formula:
\[ \text{Total triangles} = \binom{12}{3} \]
Using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
We calculate:
\[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]

3. **Calculate the Combinations of Collinear Points**:
Since 7 points are collinear, any triangle formed using these 7 points will not be valid (as they lie on a straight line). The number of ways to choose 3 points from these 7 collinear points is:
\[ \text{Collinear triangles} = \binom{7}{3} \]
Calculating this gives:
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

4. **Subtract Collinear Combinations from Total Combinations**:
To find the valid triangles, we subtract the number of invalid triangles (formed by collinear points) from the total triangles:
\[ \text{Valid triangles} = \text{Total triangles} - \text{Collinear triangles} \]
Substituting the values we calculated:
\[ \text{Valid triangles} = 220 - 35 = 185 \]

5. **Conclusion**: The number of triangles that can be formed by choosing vertices from the set of 12 points, where 7 points are collinear, is **185**.