Question

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

• A. (m + n+ 1)²
• B. mn(m + 1) (n + 1)
• C. 4^m+n-2
• D. m²n²

Answer 1

Answer: (D)

m²n²

Answer 2

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)]

= $\frac{m(1 + 2m - 1)}{2}x\frac{n(1 + 2n - 1)}{2}$= m²n².