Question

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

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

Answer 1

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

Along horizontal side one unit can be taken in $=(2m-1)$ ways
$3$ units can be taken in $=(2m-3)$ ways
$5$ units can be taken in $=(2m-5)$ ways and so on

$\therefore$ The number of ways of selecting a side horizontally is
$(2m-1)+(2m-3)+(2m-5)+...+3+1=m^2$

Similarly, the number of ways of selecting a side vertically is
$(2n-1)+(2n-3)+(2n-5)+...+3+1=n^2$
$\therefore$ Total number of rectangles $=m^2{n}^2$