Find the total number of rectangles in a square having 8 rows and 15 columns.

Options

A. 4321
B. 6320
C. 4320
D. 3456

Answer

To find the total number of rectangles in a grid with 8 rows and 15 columns, we can use the formula for the number of rectangles that can be formed in a grid. The formula is given by:

\[ \text{Total Rectangles} = \binom{m+1}{2} \times \binom{n+1}{2} \]

where \( m \) is the number of rows and \( n \) is the number of columns.

### Step 1: Identify the values of \( m \) and \( n \)
In this case:
- \( m = 8 \) (number of rows)
- \( n = 15 \) (number of columns)

### Step 2: Calculate \( m + 1 \) and \( n + 1 \)
Now, we need to calculate \( m + 1 \) and \( n + 1 \):
- \( m + 1 = 8 + 1 = 9 \)
- \( n + 1 = 15 + 1 = 16 \)

### Step 3: Calculate \( \binom{m+1}{2} \) and \( \binom{n+1}{2} \)
Next, we calculate the combinations:
\[ \binom{m+1}{2} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
\[ \binom{n+1}{2} = \binom{16}{2} = \frac{16 \times 15}{2 \times 1} = 120 \]

### Step 4: Multiply the two results
Now, we multiply the two results to find the total number of rectangles:
\[ \text{Total Rectangles} = 36 \times 120 = 4320 \]

### Final Answer
The total number of rectangles in a square having 8 rows and 15 columns is **4320**.

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