How many total rectangles are there in a standard chess board (or check board)?

Options

A. 1296
B. 512
C. 0
D. 2025

Answer

To find the total number of rectangles in a standard chessboard, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Chessboard Structure**:
A standard chessboard has 8 rows and 8 columns.

2. **Identify the Number of Lines**:
To form a rectangle, we need to select two horizontal lines and two vertical lines. In a chessboard, there are 8 rows and 8 columns, which means there are 9 horizontal lines (top and bottom of each row) and 9 vertical lines (left and right of each column).

3. **Choosing Lines to Form Rectangles**:
The number of ways to choose 2 lines from 9 lines (both horizontal and vertical) can be calculated using the combination formula:
\[ \text{Number of ways to choose 2 lines from 9} = \binom{9}{2} \]

4. **Calculate the Combinations**:
Using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
For our case:
\[ \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

5. **Total Rectangles Calculation**:
Since we need to choose 2 horizontal lines and 2 vertical lines, the total number of rectangles can be calculated as:
\[ \text{Total rectangles} = \binom{9}{2} \times \binom{9}{2} = 36 \times 36 = 1296 \]

### Final Answer:
The total number of rectangles in a standard chessboard is **1296**.

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