Question

A rectangle with sides $2m–1$ and $2n–1$ 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 length is?

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

Answer 1

The correct option is B

$m^2{n}^2$

Explanation for the correct option:

From the given figure, we have

Number of vertical lines $=1,2,3,\dots .,2m$

Number of horizontal lines $=1,2,3,\dots .,2n$

$\therefore$Required number of rectangles

$=\left[1+3+5+\dots .+(2m-1)\right]\left[(1+3+5+\dots .+(2n-1)\right]$ $=m^2{n}^2$

Hence, Option ‘B’ is Correct.