Find the total number of quadrilaterals in a rectangle having 8 rows and 15 columns.

Options

A. 4321
B. 4320
C. 1234
D. 5440

Answer

To find the total number of quadrilaterals (which are rectangles in this case) in a rectangle with 8 rows and 15 columns, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
A rectangle can be formed by choosing two horizontal lines and two vertical lines from the grid formed by the rows and columns of the rectangle.

2. **Counting Horizontal Lines**:
In a rectangle with 8 rows, there are 9 horizontal lines (one above the first row, one below the last row, and one between each pair of rows).

3. **Counting Vertical Lines**:
In a rectangle with 15 columns, there are 16 vertical lines (one to the left of the first column, one to the right of the last column, and one between each pair of columns).

4. **Choosing Lines to Form Rectangles**:
To form a rectangle, we need to choose 2 out of the 9 horizontal lines and 2 out of the 16 vertical lines. The number of ways to choose 2 lines from n lines is given by the combination formula \( C(n, 2) = \frac{n!}{2!(n-2)!} \).

5. **Calculating Combinations**:
- For horizontal lines:
\[ C(9, 2) = \frac{9 \times 8}{2 \times 1} = 36 \]
- For vertical lines:
\[ C(16, 2) = \frac{16 \times 15}{2 \times 1} = 120 \]

6. **Total Rectangles**:
The total number of rectangles (quadrilaterals) is the product of the combinations of horizontal and vertical lines:
\[ \text{Total Rectangles} = C(9, 2) \times C(16, 2) = 36 \times 120 = 4320 \]

### Final Answer:
The total number of quadrilaterals in a rectangle having 8 rows and 15 columns is **4320**.