Answer these questions based on the following informations
There are fifteen points on the circumference of a circle
Find the maximum number of triangles formed by joining the fifteen points.

Options

A. 455
B. 355
C. 475
D. 525

Answer

To find the maximum number of triangles that can be formed by joining 15 points on the circumference of a circle, we can use the concept of combinations. Specifically, we need to choose 3 points out of the 15 to form a triangle.

### Step-by-Step Solution:

1. **Understand the Problem**: We have 15 points on the circumference of a circle, and we want to find out how many triangles can be formed by selecting any 3 points from these 15.

2. **Use the Combination Formula**: The number of ways to choose \( r \) objects from \( n \) objects is given by the combination formula:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
In our case, \( n = 15 \) and \( r = 3 \).

3. **Substitute Values into the Formula**:
\[ C(15, 3) = \frac{15!}{3!(15-3)!} = \frac{15!}{3! \cdot 12!} \]

4. **Simplify the Factorials**: We can simplify \( 15! \) as follows:
\[ 15! = 15 \times 14 \times 13 \times 12! \]
Thus, the expression becomes:
\[ C(15, 3) = \frac{15 \times 14 \times 13 \times 12!}{3! \cdot 12!} \]

5. **Cancel Out \( 12! \)**: The \( 12! \) in the numerator and denominator cancels out:
\[ C(15, 3) = \frac{15 \times 14 \times 13}{3!} \]

6. **Calculate \( 3! \)**:
\[ 3! = 3 \times 2 \times 1 = 6 \]

7. **Final Calculation**:
\[ C(15, 3) = \frac{15 \times 14 \times 13}{6} \]
Now, calculate \( 15 \times 14 \times 13 \):
\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
Now divide by 6:
\[ \frac{2730}{6} = 455 \]

8. **Conclusion**: The maximum number of triangles that can be formed by joining the 15 points is \( 455 \).

### Final Answer:
The maximum number of triangles formed by joining the fifteen points is **455**.