Factorials and ⁿPᵣ

Products like 5 × 4 × 3 × 2 × 1 come up so often that they get a name. n factorial is n! = 1 × 2 × 3 × ... × n.

• 1! = 1, 3! = 6, 4! = 24, 5! = 120, 6! = 720.
• By agreement 0! = 1, which makes the formulas below work in every case.
• Each factorial contains the one before it: n! = n × (n – 1)!, so 7! = 7 × 6!.

That last fact lets you cancel before multiplying: 9!/7! = 9 × 8 = 72, and 10!/(8! × 2!) = (10 × 9)/2 = 45.

A permutation is an arrangement of objects in a definite order. The number of ways to arrange r objects chosen from n different ones, with no repetition, is

• ⁿPᵣ = n(n – 1)(n – 2)...(n – r + 1) = n!/(n – r)!: r boxes, filled with n, then n – 1, and so on.
• ⁿPₙ = n!: arranging all n objects. ⁿP₀ = 1.

Examples:

• 5 different books on a shelf: 5! = 120 orders.
• Gold, silver and bronze among 8 runners: ⁸P₃ = 8 × 7 × 6 = 336 ways. Order matters here: gold to A and silver to B is different from the other way round.
• ⁷P₃ = 7!/4! = 7 × 6 × 5 = 210.

Equations in n. If ⁿP₂ = 30, then n(n – 1) = 30 = 6 × 5, so n = 6.

With repetition, each of the r places can be any of the n objects, giving nʳ arrangements. A 4-digit PIN using digits 0–9 has 10⁴ = 10 000 possibilities.