The fundamental principle of counting

Counting by listing works for small cases but quickly gets out of hand. This chapter gives short cuts, starting with one simple idea.

Fundamental principle of counting (multiplication): if one thing can be done in m ways and, after it, a second thing in n ways, then the two together can be done in m × n ways. It extends to any number of steps.

• With 3 shirts and 4 pairs of trousers you can make 3 × 4 = 12 outfits.
• Tossing a coin and rolling a die gives 2 × 6 = 12 outcomes.

Filling places. Think of the answer as a row of boxes and fill them one at a time.

• 3-letter 'words' (any arrangement, meaningful or not) from the letters of TABLE, no repetition: 5 choices for the first box, then 4, then 3, so 5 × 4 × 3 = 60.
• With repetition allowed, every box has 5 choices: 5 × 5 × 5 = 125.
• A 3-digit lock using the digits 0–9 with no digit repeated has 10 × 9 × 8 = 720 possible codes.

Restricted places go first. How many 3-digit even numbers can be made from 1, 2, 3, 4, 5 with no repetition?

• The units digit must be 2 or 4: 2 ways. Fill it first.
• Then the hundreds digit has 4 choices left and the tens digit 3.
• Total 2 × 4 × 3 = 24.

Separate cases add. The multiplication principle is for steps done one after another ('and then'). When you choose one of several separate options ('either ... or'), add the counts instead.

• To travel from A to B by one of 3 buses or one of 2 trains, there are 3 + 2 = 5 ways.
• Signals made by hanging 2 or 3 of 4 different flags one above another: with 2 flags, 4 × 3 = 12; with 3 flags, 4 × 3 × 2 = 24. Total 12 + 24 = 36.