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

Options

A. 6543
B. 2345
C. 7275
D. 6435

Answer

To find the maximum number of octagons that can be formed by joining fifteen points on the circumference of a circle, we can use the concept of combinations. An octagon consists of 8 vertices, so we need to choose 8 points from the 15 available points.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 15 points on the circumference of a circle, and we want to form octagons (which have 8 sides).

2. **Choosing Points**: To form an octagon, we need to select 8 points from the 15 points. The number of ways to choose 8 points from 15 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of points and \( r \) is the number of points to choose.

\[ \text{Number of octagons} = \binom{15}{8} \]

3. **Using the Combination Formula**: The combination formula is defined as:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In our case, \( n = 15 \) and \( r = 8 \):

\[ \binom{15}{8} = \frac{15!}{8!(15-8)!} = \frac{15!}{8! \cdot 7!} \]

4. **Calculating Factorials**: We can simplify \( \frac{15!}{8! \cdot 7!} \) by expanding \( 15! \):

\[ \binom{15}{8} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \]

5. **Performing the Calculation**: Now we calculate the numerator and denominator:

- **Numerator**:
\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
\[ 2730 \times 12 = 32760 \]
\[ 32760 \times 11 = 360360 \]
\[ 360360 \times 10 = 3603600 \]
\[ 3603600 \times 9 = 32432400 \]

- **Denominator**:
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

6. **Final Calculation**: Now we divide the numerator by the denominator:

\[ \binom{15}{8} = \frac{32432400}{5040} = 6435 \]

### Conclusion:
The maximum number of octagons that can be formed by joining the fifteen points on the circumference of a circle is **6435**.