Question

A rectangle with sides $(2n-1)$ and $(2m-1)$ unit is divided into squares of length. The number of rectangle which can be formed with sides of odd length is

• A. $m^2{n}^2$
• B. $mn(m+1)(n+1)$
• C. $4^{m+n-1}$
• D. $Noneofthese$

Answer 1

The correct option is A $m^2{n}^2$

There are $2m$ vertical (no:$1,2,...,2m$) and $2n$ horizontal lines.To form the required rectangle we meet select two horizontal line, one even numbered and one odd and one even and similarly $2$ vartical line.The no: of rectangles is then,

${}^m{C}_1{\cdot }^m{C}_1{\cdot }^n{C}_1{\cdot }^n{C}_1=m^2\cdot n^2$