The number of ways of selecting a side horizontally is
$(2m - 1 + 2m - 3 + 2m - 5 + .... + 3 + 1)$
Similarly the number of ways along vertical side is $(2n - 1 + 2n - 3 + .... + 5 + 3 + 1)$.
$\therefore$ Total number of rectangles
$ = [1 + 3 + 5 + ..... + (2m - 1)] \times [1 + 3 + 5 + .... + (2n - 1)]$
$ = \frac{{m(1 + 2m - 1)}}{2} \times \frac{{n(1 + 2n - 1)}}{2} = {m^2}{n^2}$.
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$(A)$ $\arg (-1- i )=\frac{\pi}{4}$, where $i =\sqrt{-1}$
$(B)$ The function $f: R \rightarrow(-\pi, \pi]$, defined by $f(t)=\arg (-1+i t)$ for all $t \in R$, is continuous at all points of $R$, where $i=\sqrt{-1}$
$(C)$ For any two non-zero complex numbers $z_1$ and $z_2$, $\arg \left(\left(\frac{z_1}{z_2}\right)-\arg \left(z_1\right)+\arg \left(z_2\right)\right.$ is an integer multiple of $2 \pi$.
$(D)$ For any three given distinct complex numbers, $z_1, z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $\arg \left(\frac{\left( z - z _1\right)\left( z _2- z _3\right)}{\left( z - z _3\right)\left( z _2- z _1\right)}\right)=\pi$, lies on a straight line