A rectangle with sides of length () and () units is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is [2005]

Question

A rectangle with sides of length ( $2 m - 1$ ) and ( $2 n - 1$ ) units is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is [2005]

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(3)

If we see the blocks in terms of lines then there are $2 m$ vertical lines and $2 n$ horizontal lines. To form the required rectangle, we must select two horizontal lines, one even numbered (out of $2, 4, \dots, 2 n$ ) and one odd numbered (out of $1, 3, \dots, 2 n - 1$ ) and similarly two vertical lines.

The number of rectangles $= {}^{m}C_{1} \cdot {}^{m}C_{1} \cdot {}^{n}C_{1} \cdot {}^{n}C_{1} = m^{2} n^{2}$

Solution

Q.
A rectangle with sides of length ( $2 m - 1$ ) and ( $2 n - 1$ ) units is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is [2005]

1 ${(m + n - 1)}^{2}$

2 $4^{m + n - 1}$

3 $m^{2} n^{2}$

4 $m (m + 1) n (n + 1)$

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Solution

Q. A rectangle with sides of length (2m−1) and (2n−1) units is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is [2005]

1 (m+n-1)2

2 4m+n-1

3 m2n2