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

Options

A. 84
B. 175
C. 185
D. 201

Answer

To solve the problem of how many triangles can be formed from 12 points where 7 of them are collinear (on the same straight line), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Total Points**: We have a total of 12 points.

2. **Identify Collinear Points**: Out of these 12 points, 7 points are collinear. This means that any triangle formed using these 7 points will not be valid because a triangle cannot be formed from points that are all on the same line.

3. **Calculate Total Combinations of Points**:
To find out how many triangles can be formed from 12 points without any restrictions, we use the combination formula:
\[ \text{Total combinations} = \binom{12}{3} \]
This represents the number of ways to choose 3 points from 12.

4. **Calculate Invalid Combinations**:
Now, we need to subtract the combinations that involve the 7 collinear points. The number of ways to choose 3 points from these 7 collinear points is:
\[ \text{Invalid combinations} = \binom{7}{3} \]

5. **Perform the Combinatorial Calculations**:
- Calculate \(\binom{12}{3}\):
\[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]
- Calculate \(\binom{7}{3}\):
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

6. **Calculate Valid Triangles**:
Now, subtract the invalid combinations from the total combinations:
\[ \text{Valid triangles} = \binom{12}{3} - \binom{7}{3} = 220 - 35 = 185 \]

7. **Conclusion**: The total number of triangles that can be formed from the given points is **185**.