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

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(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

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$m^2n^2$

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(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?

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