Counting: permutations and combinations

Counting principles work out how many ways something can happen without listing them all. Multiply choices together, and use factorials, permutations and combinations.

Key facts

  • Product rule: a ways then b ways gives a × b.
  • n! = n × (n − 1) × … × 1: arrangements of n different things.
  • Permutations (order matters): nPr = n!/(n − r)!.
  • Combinations (order doesn’t matter): nCr = n!/(r!(n − r)!).

Worked examples

Example 1

How many 4-digit PINs can be made with digits 0–9 (repeats allowed)?

  1. 10 choices for each of 4 digits.
  2. 10⁴ = 10000

Answer: 10 000

Example 2

How many ways can a committee of 3 be chosen from 8 people?

  1. Order doesn’t matter.
  2. 8C3 = 8!/(3! × 5!) = 56

Answer: 56

Example 3

How many different arrangements of the letters in PAPER are there?

  1. 5 letters: 5! = 120.
  2. The two Ps can swap without changing anything. 120 ÷ 2 = 60

Answer: 60