Home
Class 12
MATHS
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)`.
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • PERMUTATIONS AND COMBINATIONS

    ARIHANT MATHS|Exercise Exercise (Subjective Type Questions)|17 Videos
  • PARABOLA

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|35 Videos
  • PROBABILITY

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|55 Videos

Similar Questions

Explore conceptually related problems

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 lengths is :

Find the areas of the rectangles whose sides are: 2 m and 70 cm.

If 2hati+hatj-3hatk and mhati+3hatj-hatk are perpendicular to each other, then find 'm' . Also find the area of the rectangle having these two vectors as sides.

The perimeter of a rectangular field is 82 m and its area is 400 m^2 . Find the dimensions of the rectangle.

Area of a square plot is 2304 m^2 . Find the side of the square plot.

Perimeter of square is 2 m more than (2)/(3) the perimeter of rectangle . If perimeter of square is 10 m . Find the dimension of the rectangle if breadth is (1)/(5) the length .

Find the length of the side of a square whose area is 441 m^2 .

Find a unit vector perpendicular to the plane determined by the points (1,-1,2),(2,0,-1) and (0,2,1)dot

A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m. Find its length and breadth.

Find the length of the side of a square whose area is 2304 m^2 .