Out of 15 points in a plane, n points are in the same straight line. 445 triangles can be formed by joining these points. What is the value of n?

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem, we need to determine the value of \( n \) such that the number of triangles formed by 15 points in a plane, where \( n \) points are collinear, is equal to 445.

### Step-by-Step Solution:

1. **Understanding the Total Points**:
We have a total of 15 points in the plane. Out of these, \( n \) points are collinear, which means they lie on the same straight line.

2. **Finding Non-Collinear Points**:
The number of non-collinear points is given by:
\[ 15 - n \]

3. **Calculating Total Triangles from 15 Points**:
The total number of triangles that can be formed from 15 points is calculated using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points and \( r \) is the number of points needed to form a triangle (which is 3):
\[ \text{Total triangles} = \binom{15}{3} = \frac{15!}{3!(15-3)!} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455 \]

4. **Calculating Triangles Formed by Collinear Points**:
The number of triangles that can be formed using the \( n \) collinear points is:
\[ \text{Collinear triangles} = \binom{n}{3} = \frac{n!}{3!(n-3)!} = \frac{n(n-1)(n-2)}{6} \]

5. **Finding Non-Collinear Triangles**:
The number of triangles formed by non-collinear points is given by:
\[ \text{Non-collinear triangles} = \text{Total triangles} - \text{Collinear triangles} \]
Substituting the values we have:
\[ \text{Non-collinear triangles} = 455 - \frac{n(n-1)(n-2)}{6} \]

6. **Setting Up the Equation**:
According to the problem, the number of non-collinear triangles is equal to 445:
\[ 455 - \frac{n(n-1)(n-2)}{6} = 445 \]

7. **Solving for \( n \)**:
Rearranging the equation gives:
\[ 455 - 445 = \frac{n(n-1)(n-2)}{6} \]
\[ 10 = \frac{n(n-1)(n-2)}{6} \]
Multiplying both sides by 6:
\[ 60 = n(n-1)(n-2) \]

8. **Finding Possible Values of \( n \)**:
Now we need to find integer values of \( n \) such that \( n(n-1)(n-2) = 60 \). Testing integer values:
- For \( n = 4 \):
\[ 4 \times 3 \times 2 = 24 \quad (\text{not a solution}) \]
- For \( n = 5 \):
\[ 5 \times 4 \times 3 = 60 \quad (\text{solution found}) \]
- For \( n = 6 \):
\[ 6 \times 5 \times 4 = 120 \quad (\text{not a solution}) \]

Thus, the value of \( n \) is \( 5 \).

### Final Answer:
\[ \boxed{5} \]