Find the number of rectangles in the below figure:

The correct answer is Option (3) → 225

Every rectangle in a grid is fixed by choosing two of the vertical lines and two of the horizontal lines as its sides. The figure is a 5 \(\times\) 5 grid, so it has 6 vertical and 6 horizontal lines.

• Ways to choose 2 of the 6 vertical lines \(\displaystyle =\frac{6\times5}{2}=15\)

• Ways to choose 2 of the 6 horizontal lines \(=15\)

• Rectangles \(=15\times15=225\) (squares included, since a square is a rectangle)

Tip: In an \(m\times n\) grid the number of rectangles is \(\displaystyle \frac{m(m+1)}{2}\times\frac{n(n+1)}{2}\); for a square grid of side \(n\) the number of squares alone is \(1^2+2^2+\dots+n^2\) (55 here, Option (4)).