If 7 points out of 12 are in the same straight line, then the number of triangles formed is

Options

A. 19
B. 158
C. 185
D. 201

Answer

To find the number of triangles that can be formed from 12 points, where 7 of those points are collinear (on the same straight line), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to form triangles using 12 points, but we have to consider that 7 of these points are collinear. A triangle cannot be formed using 3 collinear points.

2. **Total Ways to Choose 3 Points**: First, we calculate the total number of ways to choose any 3 points from the 12 points. This can be calculated using the combination formula:
\[ \text{Total ways} = \binom{12}{3} \]
The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

3. **Calculating \(\binom{12}{3}\)**:
\[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3! \cdot 9!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = \frac{1320}{6} = 220 \]

4. **Ways to Choose 3 Collinear Points**: Next, we calculate the number of ways to choose 3 points from the 7 collinear points, which cannot form a triangle:
\[ \text{Collinear ways} = \binom{7}{3} \]

5. **Calculating \(\binom{7}{3}\)**:
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3! \cdot 4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]

6. **Finding Valid Triangles**: Now, we subtract the number of ways to choose 3 collinear points from the total ways to choose 3 points:
\[ \text{Valid triangles} = \binom{12}{3} - \binom{7}{3} = 220 - 35 = 185 \]

7. **Conclusion**: Therefore, the number of triangles that can be formed from the 12 points, considering that 7 points are collinear, is:
\[ \text{Number of triangles} = 185 \]

### Final Answer:
The number of triangles formed is **185**.