There is a rectangular sheet of dimension (2m | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) rectangle is formed by choosing two horizontal lines and two vertical lines. To have odd side lengths,the distance between the chosen horizontal l

Vedclass · Accepted Answer

$m^2n^2$

Vedclass · Answer

(D) rectangle is formed by choosing two horizontal lines and two vertical lines. To have odd side lengths,the distance between the chosen horizontal lines must be odd,and the distance between the chosen vertical lines must be odd.Let the horizontal side length be $L_h$ and the vertical side length be $L_v$. We require $L_h \in \{1, 3, 5, \dots, 2m-1\}$ and $L_v \in \{1, 3, 5, \dots, 2n-1\}$.For a horizontal length $k$,the number of ways to choose two lines such that the distance is $k$ is $(2m-1) - k + 1 = 2m - k$. Summing this for all odd $k \in \{1, 3, \dots, 2m-1\}$:Number of ways $= \sum_{j=1}^{m} (2m - (2j-1)) = \sum_{j=1}^{m} (2m - 2j + 1) = 2m^2 - 2\frac{m(m+1)}{2} + m = 2m^2 - m^2 - m + m = m^2$.Similarly,the number of ways to choose vertical lines with odd distance is $n^2$.Therefore,the total number of rectangles with odd side lengths is $m^2 	imes n^2 = m^2n^2$.

There is a rectangular sheet of dimension $(2m-1) \times (2n-1)$,where $m > 0, n > 0$. It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find the number of rectangles having sides of odd unit length.

Similar Questions

Two spheres of masses $m$ and $M$ are situated in air and the gravitational force between them is $F$. The space around the masses is now filled with a liquid of specific gravity $3$. The gravitational force will now be

What is a portal system?

$A$ big water drop is formed by the combination of $n$ small water drops of equal radii. The ratio of the surface energy of $n$ drops to the surface energy of the big drop is

$A$ marble block of mass $2\, kg$ lying on ice when given a velocity of $6\, m/s$ is stopped by friction in $10\, s$. Then the coefficient of friction is-

The number of sigma $(\sigma)$ and pi $(\pi)$ bonds present in benzene respectively are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module