Question

A rectangle with sides 2m -1 and 2n -1 is divided into squares of unit length by drawing parallel lines as shows in the diagram, then the number of rectangles possible with odd side lengths 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 $m^2{n}^2$

If we see the blocks in terms of lines, then there are 2m vertical lines and 2n horizontal lines.
To form the required rectangle with odd side length we must select two horizontal lines, one even numbered (out of 2, 4, ....2n) and one odd numbered (out of 1, 3, ...., 2n - 1) and similarly two vertical lines.
The number of rectangle is ${}^n{C}_1{\times }^n{C}_1{\times }^n{C}_1{\times }^n{C}_1=m^2{n}^2$