question_answer

There is a rectangular sheet of dimension \[(2m-1)\]×\[(2n-1)\], (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 [IIT Screening 2005]

A) \[{{(m+n+1)}^{2}}\]

B) \[mn(m+1)\,(n+1)\]

C) \[{{4}^{m+n-2}}\]

D) \[{{m}^{2}}{{n}^{2}}\]

Correct Answer: D

Solution :

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 \[=[1+3+5+.....+(2m-1)]\times [1+3+5+....+(2n-1)]\] \[=\frac{m(1+2m-1)}{2}\times \frac{n(1+2n-1)}{2}={{m}^{2}}{{n}^{2}}\].