A rectangle with side lengths 2m - 1 and 2n - | vedclass

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

Question

(B) rectangle is formed by choosing two horizontal lines and two vertical lines. Let the horizontal lines be $x_1, x_2, \dots, x_{2m}$ and vertical li

Vedclass · Accepted Answer

$m^2n^2$

Vedclass · Answer

(B) rectangle is formed by choosing two horizontal lines and two vertical lines. Let the horizontal lines be $x_1, x_2, \dots, x_{2m}$ and vertical lines be $y_1, y_2, \dots, y_{2n}$.For the side length of the rectangle to be odd,we must choose two lines such that the difference between their indices is odd.Let the indices of the two chosen lines be $i$ and $j$ $(i In the set of indices ${1, 2, \dots, 2m}$,there are $m$ odd numbers and $m$ even numbers.The number of ways to choose one odd and one even index is $m 	imes m = m^2$.Similarly,for the other side with $2n$ lines,the number of ways to choose two lines such that the distance is odd is $n 	imes n = n^2$.Since the choice of horizontal and vertical sides are independent,the total number of such rectangles is $m^2 	imes n^2 = m^2n^2$.

$A$ rectangle with side lengths $2m - 1$ and $2n - 1$ is divided into unit squares by drawing parallel lines as shown. How many rectangles are there such that both side lengths are odd?

Similar Questions

$A$ simple graph contains $24$ edges. The degree of each vertex is $3$. The number of vertices is:

There are $m$ points on a straight line $AB$ and $n$ points on another line $AC$,none of them being the point $A$. Triangles are formed from these points as vertices when $(i)$ $A$ is excluded $(ii)$ $A$ is included. Then the ratio of the number of triangles in the two cases is

What is the number of diagonals in an octagon?

Suppose $t_n$ is the number of triangles formed using the vertices of a regular polygon of $n$ sides. If $t_{n+1} = t_n + 28$,then $n =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module