The number of rectangles excluding squares from a reactangle of size of `12 xx 8` is :

Options

A. 1234
B. 625
C. 2460
D. 256

Answer

To find the number of rectangles excluding squares in a rectangle of size 12 by 8, we can follow these steps:

### Step 1: Calculate the Total Number of Rectangles
To calculate the total number of rectangles in a rectangle of size 12 by 8, we need to determine how many vertical and horizontal lines can be drawn.

1. **Vertical Lines**: For a rectangle of width 12, there are 13 vertical lines (0 to 12).
2. **Horizontal Lines**: For a rectangle of height 8, there are 9 horizontal lines (0 to 8).

To form a rectangle, we need to choose 2 vertical lines and 2 horizontal lines.

The number of ways to choose 2 vertical lines from 13 is given by the combination formula:
\[ \text{Number of ways to choose 2 vertical lines} = \binom{13}{2} = \frac{13 \times 12}{2 \times 1} = 78 \]

The number of ways to choose 2 horizontal lines from 9 is:
\[ \text{Number of ways to choose 2 horizontal lines} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]

Now, multiply the two results to get the total number of rectangles:
\[ \text{Total Rectangles} = 78 \times 36 = 2808 \]

### Step 2: Calculate the Total Number of Squares
Next, we need to find the total number of squares that can be formed in the rectangle.

1. **1x1 Squares**: The number of 1x1 squares is \(12 \times 8 = 96\).
2. **2x2 Squares**: The number of 2x2 squares is \(11 \times 7 = 77\).
3. **3x3 Squares**: The number of 3x3 squares is \(10 \times 6 = 60\).
4. **4x4 Squares**: The number of 4x4 squares is \(9 \times 5 = 45\).
5. **5x5 Squares**: The number of 5x5 squares is \(8 \times 4 = 32\).
6. **6x6 Squares**: The number of 6x6 squares is \(7 \times 3 = 21\).
7. **7x7 Squares**: The number of 7x7 squares is \(6 \times 2 = 12\).
8. **8x8 Squares**: The number of 8x8 squares is \(5 \times 1 = 5\).

Now, sum all the squares:
\[ \text{Total Squares} = 96 + 77 + 60 + 45 + 32 + 21 + 12 + 5 = 348 \]

### Step 3: Calculate the Number of Rectangles Excluding Squares
Finally, we subtract the total number of squares from the total number of rectangles:
\[ \text{Rectangles Excluding Squares} = \text{Total Rectangles} - \text{Total Squares} = 2808 - 348 = 2460 \]

Thus, the number of rectangles excluding squares in a rectangle of size 12 by 8 is **2460**.

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