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Maths-

General

Easy

Question

A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length –

$m^{2} \cdot - n^{2}$
m(m + 1) n(n + 1)
$4 m + n - 2$

m²n²

hint Hint:

$T h e r e space w i l l space b e space 2 m space space v e r t i c a l space l i n e s space a n d space 2 n space space h o r i z o n t a l space l i n e s. space T o space f o r m space a space r e c tan g l e comma space w e space n e e d space t w o space l i n e s space f r o m space e a c h space s e t. space B u t space t h e space s i d e s space n e e d space t o space b e space o d d space n u m b e r e d comma space s o space w e space w i l l space s e l e c t space o n e space o d d space a n d space o n e space e v e n space n u m b e r e d space s i d e space f r o m space b o t h space s e t s. space A n d space t h e n space u sin g space t h e space c o n c e p t space o f space p e r m u t a t i o n space w e space w i l l space f i n d space t h e space n u m b e r space o f space r e c tan g l e s space p o s s i b l e space w i t h space o d d space s i d e space l e n g t h.$

The correct answer is: m²n²

$W e space h a v e comma space n u m b e r space o f space w a y s space o f space s e l e c t i n g space v e r t i c a l space s i d e s space equals 1 plus 3 plus 5 plus.... plus left parenthesis 2 m minus 1 right parenthesis space space w h i c h space i s space e q u a l space t o space m squared space.$
$S i m i l a r l y comma space t h e space n u m b e r space o f space w a y s space o f space s e l e c t i n g space h o r i z o n t a l space s i d e s space equals 1 plus 3 plus 5 plus.... plus left parenthesis 2 n minus 1 right parenthesis space space w h i c h space i s space e q u a l space t o space n squared space.$

$S o comma space t h e space n u m b e r space o f space r e c tan g l e s space p o s s i b l e space w i t h space o d d space s i d e space l e n g t h space equals space space n u m b e r space o f space w a y s space o f space s e l e c t i n g space v e r t i c a l space s i d e s space cross times space space n u m b e r space o f space w a y s space o f space s e l e c t i n g space h o r i z o n t a l space s i d e s. T h e r e f o r e comma space w e space g e t space t h e space n u m b e r space o f space r e c tan g l e s space p o s s i b l e space w i t h space o d d space s i d e space l e n g t h space equals m squared n squared space.$

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