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

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) m2n2 (B) mn(m + 1)(.n+1.) (C) 4m + n - 1 (D) None of these

Tardigrade Admin · Accepted Answer

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 + \dots .. + (2 m - 1)) (1 + 3 + 5 + \dots .. + (2 n - 1)) = m^{2} n^{2}

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

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

$mn (m + 1) (n + 1)$

$4^{m + n - 1}$

None of these

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

Q. 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

m2n2

mn(m+1)(n+1)

4m+n−1