Question

A rectangle with sides $2m-1 \, $ and $2n-1$ is divided into squares of unit length. The number of rectangles that can be formed with both the sides of odd length is

• A. $m^{2}n^{2}$
• B. $mn\left(m + 1\right)\left(\right.n+1\left.\right)$
• C. $4^{m + n - 1}$
• D. None of these

Answer 1

Answer: (A)

Number of rectangles possible is
= (Number of ways to choose a horizontal side of odd length).(Number of ways to choose a vertical side of odd length)
$(1+3+5+\ldots . .+(2 m-1))(1+3+5+\ldots . .+(2 n-1))=m^{2} n^{2}$