Counting & combining

You're standing in front of your wardrobe: 3 shirts, 2 pairs of trousers. How many different outfits is that? Or you have to pick a PIN and wonder how many there even are. Combinatorics answers questions like these, the art of clever counting. The best part: almost all you need is multiplication.

The counting principle

With each of the 3 shirts you can wear either of the 2 pairs of trousers. That makes 3 · 2 = 6 outfits. This is the counting principle: when you make several choices one after another, you multiply the number of options for each choice. Add a choice between 2 pairs of shoes and you get 3 · 2 · 2 = 12 outfits.

The tree diagram

If you want a picture of the counting principle, draw a tree: 3 branches leave the start, one per shirt. From each shirt branch, 2 branches split off, one per pair of trousers. At the end of the tree you count the tips: 6 of them, one outfit per path. The tree diagram shows every possibility as its own path from left to right.

Orders and PIN codes

If 3 people sit down on 3 seats, the first person can choose from 3 seats, the second from 2, the last from only 1: that is 3 · 2 · 1 = 6 orders. A 4-digit PIN works differently: every position can be any of the 10 digits, repeats allowed. That makes 10 · 10 · 10 · 10 = 10000 codes. So always check whether an option is used up after being chosen or stays available.