Question

A rectangle with sides $2m-1,2n-1$ is divided into squares of unit length by drawing parallel lines as shown in the diagram. The number of rectangles with odd side length is

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

Answer 1

The correct option is C $m^2{n}^2$

The number of vertical lines is $2m$ and the number of horizontal lines is $2n$.

Now, we select $2$ lines each from both the sets.

But, as the sides need to be odd numbered, we need to select 1 odd numbered and 1 even numbered side from both sets.

Hence, the required answer is ${}^m{C}_1^m{C}_1^n{C}_1^n{C}_1=m^2{n}^2$ as half sides are odd numbered and the other half are even numbered.