The Counting Principle

If you have 4 shirts and 3 pairs of trousers, you have 4 × 3 = 12 outfits. Each shirt pairs with each pair of trousers.

That is the fundamental counting principle: multiply the number of options at each stage.

It extends to as many stages as you like. A 4-digit PIN using digits 0-9 has 10 × 10 × 10 × 10 = 10,000 possibilities.

If repeats are not allowed, the count shrinks at each stage. Arranging 5 people in a row: 5 × 4 × 3 × 2 × 1 = 120.

The multiplication principle states that if a procedure has k independent stages with n₁, n₂, …, n_k options, the total number of outcomes is the product n₁·n₂·…·n_k. Independence here means the number of options at each stage does not depend on earlier choices, though the specific options may.

Two standard specialisations follow. Permutations count ordered arrangements without repetition: P(n,r) = n!/(n−r)!. Combinations count unordered selections: C(n,r) = n!/(r!(n−r)!), dividing by r! precisely because each unordered set was counted once for every ordering of its elements.

Deciding between them is a single question: would swapping two chosen items give a different outcome? Seating arrangements and passwords say yes (permutation); committees and lottery tickets say no (combination). These are developed further with the binomial theorem in Algebra 2.