A rectangle with sides `2m-1a n d2n-1` is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is fig a.`(m+n-1)^2` b. `4^(m+n-1)` c. `m^2n^2` d. `m(m+1)n(n+1)`

Options

A. `m^(2)n^(2)`
B. `mn(m+1)(n+1)`
C. `4^(m+n-1)`
D. none of these

Explanation

Clearly,
Number of horizontal rectangles of 1 units length `=(2m-1)`
Number of horizontal rectangles of 3 unit length `=(2m-3)`
Number of horizontal rectangles of 5 unit length `=(2m-5)`
Similarly,
Number of rectangles of `(2m-1)` unit length =1
`:.` Total number of horizontal rectangles of odd length
`=(2m-1)+(2m-3)+....+3+1=m^(2)`

Similarly
Total number vertical rectangles of odd length `=n^(2)`
`:.` Total numbr of rectangles `=m^(2)xxn^(2)`