The number of rectangles excluding squares from a rectangle of `9 times 6` size is

Options

A. 842
B. 791
C. 391
D. none of these

Answer

To solve the problem of finding the number of rectangles excluding squares from a rectangle of size 9 by 6, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Number of Lines**:
- For a rectangle of dimensions 9 (length) by 6 (width), we need to find the number of horizontal and vertical lines.
- The number of horizontal lines = height + 1 = 6 + 1 = 7.
- The number of vertical lines = width + 1 = 9 + 1 = 10.

2. **Calculate Total Rectangles**:
- The total number of rectangles that can be formed is given by selecting 2 horizontal lines and 2 vertical lines.
- The formula for selecting 2 lines from n lines is given by \( \binom{n}{2} \).
- Thus, the total rectangles = \( \binom{10}{2} \times \binom{7}{2} \).
- Calculate \( \binom{10}{2} = \frac{10 \times 9}{2} = 45 \).
- Calculate \( \binom{7}{2} = \frac{7 \times 6}{2} = 21 \).
- Therefore, total rectangles = \( 45 \times 21 = 945 \).

3. **Calculate Total Squares**:
- Next, we need to calculate the number of squares that can be formed within the rectangle.
- The size of the squares can range from 1x1 to 6x6 (the smaller dimension of the rectangle).
- For each size \( k \times k \), the number of such squares is given by \( (9 - k + 1) \times (6 - k + 1) \).

- Calculate for each square size:
- For 1x1 squares: \( 9 \times 6 = 54 \)
- For 2x2 squares: \( 8 \times 5 = 40 \)
- For 3x3 squares: \( 7 \times 4 = 28 \)
- For 4x4 squares: \( 6 \times 3 = 18 \)
- For 5x5 squares: \( 5 \times 2 = 10 \)
- For 6x6 squares: \( 4 \times 1 = 4 \)

- Total squares = \( 54 + 40 + 28 + 18 + 10 + 4 = 154 \).

4. **Calculate Rectangles Excluding Squares**:
- Finally, to find the number of rectangles excluding squares, we subtract the total number of squares from the total number of rectangles.
- Rectangles excluding squares = Total rectangles - Total squares = \( 945 - 154 = 791 \).

### Final Answer:
The number of rectangles excluding squares from a rectangle of size 9 by 6 is **791**.