There is a rectangular sheet of dimension (2m-1 ) (2n-1 ), (where…

MCQ

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

A
$\big(\text{m+n+1}\big)^2$
B
$\text{mn}\big(\text{m+1}\big)\big(\text{n+1}\big)$
C
$4^\text{m+n-2}$
✓
$\text{m}^2\text{n}^2$

✓

Answer

Correct option: D.

$\text{m}^2\text{n}^2$

Total no. of horizontal line $= 2m$ Total no. of vertical lines $= 2n$ $($$\because$ Each line is at unit distance and hence, total no. of lines = Distance/lenght $+1).$
To form a square from three lines,we mustselect one even and one odd numbered horizontal and vertical line
$\therefore$ Ways possible of selecting such squares $=(\text{c}_{1}^\text{m})\times(\text{c}_{1}^\text{m})\times$ $(\text{c}_{1}^\text{n}\times\text{c}_{1}^\text{n})=$ $\text{c}_{1}^\text{m}\times\text{c}_{1}^\text{m}\times\text{c}_{1}^\text{n}\times\text{c}_{1}^\text{n}=$ $\text{m}^2\times\text{n}^2=\text{m}^2\text{n}^2$

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