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There is a rectangular sheet of dimensio...

There is a rectangular sheet of dimension `(2m-1)xx(2n-1)`, (where `m > 0, n > 0`) It has been divided into square of unit area by drawing line perpendicular to the sides. Find the number of rectangles having sides of odd unit length.

A

`(m+n+1)^(2)`

B

`mn(m+1)(n+1)`

C

`m^(m+n-2)`

D

`m^(2)n^(2)`

Text Solution

Verified by Experts

The correct Answer is:
D
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)`.
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Knowledge Check

  • There is a rectangular sheet of dimensions ( 2m-1) times (2n-1) (where m gt 0, n gt 0) It has been divided into squares 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
    `m^2n^2`

  • A rectangle with sides (2m - 1) and (2n - 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 length is

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

  • A portion inside a rectangle of length 5 m and breadth 2 m is shaded in the form of square of side 2 m. What is the ratio of the area of the shaded square to the unshaded portion of the rectangle ?

    A
    `3 :2`
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    C
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