Question

A rectangle with sides 2m - 1 and 2n - 1 is divided into square of unit length by drawing parallel lines as shown in diagram, then the number of rectangles possible with odd side length is

• A. $(m+n-1{)}^2$
• B. $4^{m+n-1}$
• C. $m^2-n^2$
• D. $m(m+1)n(n+1)$

Answer 1

The correct option is C $m^2-n^2$

REF.Image

A rectangle with sides 2m-1 and 2n-1 is divide into square units length by drawing parallel lines as shown in diagram, then the number of rectangle possible wide odd side length

Sol:

There are 2m vertical (number 1,2........2m) and 2n horizontal lines (numbered 1,2.......2n).

To form the required rectangle we must selected two horizontal lines, one even numbered and one odd numbered and similarly two vertical lines.

The number of rectangle is then

$(1+3+5+......+(2m-1\left)\right)(1+3+5+......+(2n-1\left)\right)-m^2.n^2$

$\therefore$ so, the answer is $c.m^2-n^2$