The number of rectangle , of any sixe excluding squares from the rectangle of size 8 `xx` 7 is

Options

A. (a)784
B. (b)840
C. (c)896
D. (d)None of these

Answer

To solve the problem of finding the number of rectangles of any size excluding squares from a rectangle of size 8 x 7, we can follow these steps:

### Step 1: Calculate the total number of rectangles in an 8 x 7 rectangle.
To find the total number of rectangles, we can use the formula for the number of rectangles that can be formed in a grid. The number of rectangles is given by the formula:

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

where \( m \) is the number of rows and \( n \) is the number of columns. Here, \( m = 8 \) and \( n = 7 \).

Calculating this gives:

\[ \text{Total Rectangles} = \binom{9}{2} \times \binom{8}{2} \]

### Step 2: Calculate \( \binom{9}{2} \) and \( \binom{8}{2} \).
Using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):

\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

### Step 3: Multiply the two results to get the total number of rectangles.
Now we can multiply the two results:

\[ \text{Total Rectangles} = 36 \times 28 = 1008 \]

### Step 4: Calculate the number of squares in the rectangle.
Next, we need to find the number of squares that can be formed in the rectangle. The number of squares of size \( k \times k \) that can fit in an \( m \times n \) rectangle is given by:

\[ \text{Squares} = (m-k+1)(n-k+1) \]

We will sum this for all possible square sizes \( k \) from 1 to the minimum of \( m \) and \( n \) (which is 7 in this case).

Calculating the number of squares for each size:

- For \( k = 1 \): \( 8 \times 7 = 56 \)
- For \( k = 2 \): \( 7 \times 6 = 42 \)
- For \( k = 3 \): \( 6 \times 5 = 30 \)
- For \( k = 4 \): \( 5 \times 4 = 20 \)
- For \( k = 5 \): \( 4 \times 3 = 12 \)
- For \( k = 6 \): \( 3 \times 2 = 6 \)
- For \( k = 7 \): \( 2 \times 1 = 2 \)

### Step 5: Sum the number of squares.
Now, we sum these values:

\[ \text{Total Squares} = 56 + 42 + 30 + 20 + 12 + 6 + 2 = 168 \]

### Step 6: Subtract the number of squares from the total number of rectangles.
Finally, we subtract the total number of squares from the total number of rectangles to find the number of rectangles that are not squares:

\[ \text{Rectangles excluding squares} = 1008 - 168 = 840 \]

### Final Answer:
The number of rectangles of any size excluding squares from the rectangle of size 8 x 7 is **840**.

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