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

Options

A. (a) 365
B. (b) 1035
C. (c) 1155
D. (d) 1365

Answer

To find the maximum number of quadrilaterals that can be formed by joining 15 points on the circumference of a circle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 15 points on the circumference of a circle. A quadrilateral is formed by joining 4 points. Since all points are on the circumference, they are non-collinear, which means any combination of 4 points will form a quadrilateral.

2. **Using Combinations**:
To find the number of ways to choose 4 points from 15, we use the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of points (15 in this case) and \( r \) is the number of points to choose (4 for a quadrilateral).

3. **Applying the Formula**:
Here, we need to calculate \( 15C4 \):
\[ 15C4 = \frac{15!}{4!(15-4)!} = \frac{15!}{4! \cdot 11!} \]

4. **Simplifying the Factorials**:
We can simplify \( 15! \) as follows:
\[ 15! = 15 \times 14 \times 13 \times 12 \times 11! \]
Thus, we have:
\[ 15C4 = \frac{15 \times 14 \times 13 \times 12 \times 11!}{4! \times 11!} \]
The \( 11! \) cancels out:
\[ 15C4 = \frac{15 \times 14 \times 13 \times 12}{4!} \]

5. **Calculating \( 4! \)**:
We know that:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

6. **Final Calculation**:
Now substituting \( 4! \) back into the equation:
\[ 15C4 = \frac{15 \times 14 \times 13 \times 12}{24} \]
Let's calculate the numerator:
\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
\[ 2730 \times 12 = 32760 \]
Now divide by 24:
\[ 15C4 = \frac{32760}{24} = 1365 \]

7. **Conclusion**:
Therefore, the maximum number of quadrilaterals that can be formed by joining the 15 points is **1365**.