There is a rectangular sheet of dimensions (`2m-1) times (2n-1)` (where `m gt 0, n gt 0)` It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find number of rectangles having sides of odd unit length.

Options

A. `(m+n+1)^2`
B. `mn(m+1)(n+1)`
C. `4^(m+n-2)`
D. `m^2n^2`

Answer

To find the number of rectangles with sides of odd unit length in a rectangular sheet of dimensions \((2m-1) \times (2n-1)\), we can follow these steps:

### Step 1: Determine the number of horizontal and vertical lines
The dimensions of the rectangle are \(2m - 1\) and \(2n - 1\). Since the rectangle is divided into unit squares, the number of horizontal lines is equal to the height plus one, and the number of vertical lines is equal to the width plus one.

- **Horizontal lines**: \(2m\) (from 0 to \(2m-1\))
- **Vertical lines**: \(2n\) (from 0 to \(2n-1\))

### Step 2: Identify the odd-length sides
To form rectangles with odd-length sides, we need to select lines such that the distance between them is odd. This can be achieved by selecting lines at odd indices.

### Step 3: Count the odd-indexed lines
In the \(2m\) horizontal lines, the odd-indexed lines are:
- 1, 3, 5, ..., \(2m-1\) (which gives us \(m\) odd lines)

In the \(2n\) vertical lines, the odd-indexed lines are:
- 1, 3, 5, ..., \(2n-1\) (which gives us \(n\) odd lines)

### Step 4: Calculate the number of ways to select lines
To form a rectangle, we need to select two horizontal lines and two vertical lines. The number of ways to select 2 lines from \(m\) odd horizontal lines is given by \(\binom{m}{2}\), and the number of ways to select 2 lines from \(n\) odd vertical lines is given by \(\binom{n}{2}\).

The total number of rectangles with odd-length sides can be calculated as:
\[ \text{Total rectangles} = \binom{m}{2} \times \binom{n}{2} \]

### Step 5: Simplify the combinations
Using the formula for combinations:
\[ \binom{m}{2} = \frac{m(m-1)}{2}, \quad \binom{n}{2} = \frac{n(n-1)}{2} \]
Thus,
\[ \text{Total rectangles} = \frac{m(m-1)}{2} \times \frac{n(n-1)}{2} = \frac{m(m-1)n(n-1)}{4} \]

### Step 6: Final Result
The number of rectangles having sides of odd unit length is:
\[ \frac{m(m-1)n(n-1)}{4} \]