Find the total number of rectangles in a square having 10 rows and 10 columns

Options

A. 3225
B. 1625
C. 1044
D. 3025

Answer

To find the total number of rectangles in a square grid with 10 rows and 10 columns, we can follow these steps:

### Step 1: Understand the Structure of Rectangles
A rectangle in a grid is defined by selecting two horizontal lines and two vertical lines. In a 10x10 grid, there are 10 rows and 10 columns, which means there are 11 horizontal lines (including the top and bottom) and 11 vertical lines (including the left and right sides).

### Step 2: Calculate the Number of Ways to Choose Lines
To form a rectangle, we need to choose 2 lines from the 11 horizontal lines and 2 lines from the 11 vertical lines. The number of ways to choose 2 lines from 11 can be calculated using the combination formula:

\[ \text{Number of ways to choose 2 lines from 11} = \binom{11}{2} \]

### Step 3: Calculate \(\binom{11}{2}\)
The combination formula \(\binom{n}{r}\) is given by:

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

For our case:

\[ \binom{11}{2} = \frac{11!}{2!(11-2)!} = \frac{11 \times 10}{2 \times 1} = 55 \]

### Step 4: Calculate Total Rectangles
Since we need to choose 2 horizontal lines and 2 vertical lines, the total number of rectangles can be calculated by multiplying the number of ways to choose horizontal lines and vertical lines:

\[ \text{Total rectangles} = \binom{11}{2} \times \binom{11}{2} = 55 \times 55 = 3025 \]

### Conclusion
Thus, the total number of rectangles in a square grid with 10 rows and 10 columns is **3025**.

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