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Double counting, also known as “counting in two ways,” is a clever technique used in combinatorics (the mathematics of counting). It’s a method for proving that two expressions are equal by showing that they both count the same set of objects, but in different ways.
Simple Example: Arranging Objects in a Grid
Imagine you have a grid of dots with 3 rows and 4 columns. You want to find the total number of dots.
- Method 1: Count the dots in each row and add them up. Since there are 4 dots in each row, the total is 3 rows * 4 dots/row = 12 dots.
- Method 2: Count the dots in each column and add them up. Since there are 3 dots in each column, the total is 4 columns * 3 dots/column = 12 dots.