Permutations And Combinations Question 204

Question: 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]

Options:

A) $ {{(m+n+1)}^{2}} $

B) $ mn(m+1)(n+1) $

C) $ {4^{m+n-2}} $

D) $ m^{2}n^{2} $

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Answer:

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} $ .