There is a rectangular sheet of dimension (2m – 1) x (2n – 1), (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
Text Solution
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Along horizontal side one unit can be taken in (2m – 1) ways and 3 unit side can be taken in 2m – 3 ways.
∴ ∴ 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)
∴ ∴ Total number of rectangles
= [1 + 3 + 5 + ..........+ (2m – 1)] x [1 + 3 + 5 + .........+
(2n – 1)]
= 
= m 2 n 2 .
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