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 with 12 points where 7 of them are collinear, we can follow these steps:

### Step-by-Step Solution:

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

2. **Determine Points Needed for a Triangle**:
A triangle is formed by selecting 3 points.

3. **Calculate Total Combinations of 3 Points from 12**:
The total number of ways to choose 3 points from 12 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \(n = 12\) and \(r = 3\):
\[ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]

4. **Calculate Combinations of 3 Points from the 7 Collinear Points**:
Since 7 points are collinear, any triangle formed by choosing 3 points from these 7 will not be valid (as they lie on the same straight line). We need to subtract these combinations:
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

5. **Calculate Valid Triangles**:
Now, we subtract the number of invalid triangles (those formed by the 7 collinear points) from the total combinations:
\[ \text{Valid triangles} = \binom{12}{3} - \binom{7}{3} = 220 - 35 = 185 \]

6. **Final Answer**:
Therefore, the number of triangles that can be formed is **185**.