Counting rectangles in a grid
The question
A pegboard holds its pegs in a rectangle 3 pegs across and 4 pegs down — 12 pegs altogether. A rectangle is any four pegs that make its corners, sides along the rows and columns.How many rectangles are there?Why
Count the rectangles without drawing them
How it works.
across:3 down:4
columns:3 rows:6
3×6=18
18
- 01
Start from the pegs: 3 across and 4 down. A rectangle with sides on the grid needs two of the columns for its left and right edges, and two of the rows for its top and bottom.
- 02
Choosing 2 of the 3 columns is C(3,2) = 3, and 2 of the 4 rows is C(4,2) = 6.
- 03
Every pair of columns goes with every pair of rows, so multiply them: 3 × 6 = 18.
- 04
So 18 rectangles — not the 6 little unit cells, which are only the smallest ones.