Find the total number of rectangles when 8 vertical lines intersect 15 horizontal lines such that all the vertical lines are parallel and equidistant to each other, and all the horizontal lines are also parallel and equidistant to each other.

Options

A. 3996
B. 6040
C. 2940
D. 6734

Answer

To find the total number of rectangles formed by 8 vertical lines and 15 horizontal lines, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 8 vertical lines and 15 horizontal lines. To form a rectangle, we need to select 2 vertical lines and 2 horizontal lines.

2. **Selecting Vertical Lines**:
The number of ways to select 2 lines from 8 vertical lines can be calculated using the combination formula \( nCk \), which is given by:
\[ nCk = \frac{n!}{k!(n-k)!} \]
Here, \( n = 8 \) and \( k = 2 \):
\[ 8C2 = \frac{8!}{2!(8-2)!} = \frac{8!}{2! \cdot 6!} \]
Simplifying this:
\[ 8C2 = \frac{8 \times 7}{2 \times 1} = 28 \]

3. **Selecting Horizontal Lines**:
Similarly, the number of ways to select 2 lines from 15 horizontal lines is:
\[ 15C2 = \frac{15!}{2!(15-2)!} = \frac{15!}{2! \cdot 13!} \]
Simplifying this:
\[ 15C2 = \frac{15 \times 14}{2 \times 1} = 105 \]

4. **Calculating Total Rectangles**:
The total number of rectangles can be found by multiplying the number of ways to choose the vertical lines by the number of ways to choose the horizontal lines:
\[ \text{Total Rectangles} = 8C2 \times 15C2 = 28 \times 105 \]

5. **Performing the Multiplication**:
Now, we calculate:
\[ 28 \times 105 = 2940 \]

6. **Conclusion**:
Therefore, the total number of rectangles formed by the intersection of 8 vertical lines and 15 horizontal lines is **2940**.