There are 15 points in a plane, no three of which are in a straight line except 4, all of which are in a straight line. The number of triangles that can be formed by using these 15 points is:

Options

A. 404
B. 415
C. 451
D. 490

Answer

To solve the problem of finding the number of triangles that can be formed using 15 points in a plane, where no three points are collinear except for 4 points that are collinear, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 15 points in total. Out of these, 4 points are collinear (i.e., they lie on the same straight line), and the remaining 11 points are not collinear with each other or with the 4 points.

2. **Finding Total Triangles from 15 Points**:
To find the total number of triangles that can be formed from 15 points, we use the combination formula \( C(n, r) \), which gives us the number of ways to choose \( r \) items from \( n \) items without regard to the order of selection. The formula is:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
For our case, we need to choose 3 points from 15:
\[ C(15, 3) = \frac{15!}{3!(15-3)!} = \frac{15!}{3! \cdot 12!} \]

3. **Calculating \( C(15, 3) \)**:
Simplifying \( C(15, 3) \):
\[ C(15, 3) = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = \frac{2730}{6} = 455 \]

4. **Finding Triangles from Collinear Points**:
Since the 4 collinear points cannot form a triangle, we calculate the number of triangles that can be formed using these 4 points:
\[ C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = 4 \]

5. **Calculating the Valid Triangles**:
Now, we subtract the number of triangles that can be formed from the 4 collinear points from the total number of triangles:
\[ \text{Valid triangles} = C(15, 3) - C(4, 3) = 455 - 4 = 451 \]

6. **Final Answer**:
Therefore, the number of triangles that can be formed using the 15 points is:
\[ \boxed{451} \]