A rectangle with sides of lengths (2 n-1) and (2 m-1) units is divided into squares of unit lengths. The number of rectangles which can be formed with sides of odd length is

Options

A. `(m+n+1)^(2)`
B. `mn(m+1)(n+1)`
C. `m^(m+n-2)`
D. `m^(2)n^(2)`

Answer

D

Explanation

Along horizontal side one unit can be taken in `(2m-1)` ways and 3 unit side cann be taken in (2m-3) ways. The number of ways of selecting a side horizontally is
`(2m-1+2m-3+2m-5+ . . .+3+1)=(m)/(2)(2m-1+1)=m^(2)`

similarly, the numbe rof ways along vertical side is
`(2n-1+2n-3+ . .+5+3+1)=(n)/(2)(2n-1+1)=n^(2)`
`therefore`Total number of rectangles`=m^(2)n^(2)`.