Cambridge (CIE) IGCSE · Additional Maths
Permutations & Combinations
The big idea: when order matters, you're counting permutations —
when order doesn't matter, you're counting combinations. Almost every question in this
chapter is really just asking: "does order matter here?"
Summary — What This Chapter Covers
- Arrangements: the number of ways to order n different objects is
n! (n factorial).
- Factorials:
n! = n × (n-1) × (n-2) × ... × 2 × 1, with the special rule 0! = 1.
- Permutations (order matters): arranging r items out of n different items — formula
ⁿPᵣ = n!/(n−r)!
- Combinations (order doesn't matter): choosing r items out of n — formula
ⁿCᵣ = n!/[(n−r)! r!]
- Restrictions: items that must stay together, must be separated, or must sit in fixed positions each have their own trick.
- AND vs OR: multiply when you need one thing and another; add when it's one thing or another.
1. Arrangements & Factorials
Why does n × (n−1) × (n−2) × ... work?
Imagine you have 3 letters — A, B, C — and 3 empty seats in a row. Think about filling the seats one at a time:
- Seat 1: any of the 3 letters could sit here → 3 choices
- Seat 2: one letter is already used, so only 2 remain → 2 choices
- Seat 3: only 1 letter is left → 1 choice
Multiply the choices at each stage: 3 × 2 × 1 = 6. That's every possible order:
ABC, ACB, BAC, BCA, CAB, CBA. This "count the options at each position, then multiply" logic is the
engine behind this entire chapter — everything else is a variation on it.
What exactly is a factorial?
"Factorial" is just a mathematical operation — the same category as +, −, ×, ÷ — written with an exclamation mark.
5! is read "5 factorial" and means 5 × 4 × 3 × 2 × 1 = 120.
Two special facts worth knowing cold:
- Negative factorials don't exist — you cannot arrange −3 objects, so it's undefined.
- 0! = 1 — this feels weird at first, but think about it: there are no positive integers below zero, so "arranging zero items" can only be done in exactly one way (do nothing). This isn't just a convention — it's what keeps all the formulas in this chapter working correctly.
Good to Know
Most standard calculators can't handle factorials above about 70! — the number becomes too enormous. Try your own calculator and find the biggest x where x! still works.
Worked Example
Find how many distinct arrangements there are of the letters in the word MATHS.
MATHS has 5 different letters, so there are 5 options for the first space, 4 for the second, 3 for the third, and so on:
5 × 4 × 3 × 2 × 1 = 120 distinct arrangements
Simplifying factorial fractions
Big factorials divided by smaller ones cancel out beautifully. Think of 7!/4! written out in full:
This gives us two handy identities worth memorising:
Worked Example
(i) Show, by writing 8! and 5! in full and cancelling, that 8!/5! = 8 × 7 × 6.
(ii) Hence, simplify n!/(n−3)!
(i) Write both out fully:
5! = 5×4×3×2×1 8! = 8×7×6×5×4×3×2×1
8 × 7 × 6
(ii)
n!/(n−3)! = n(n−1)(n−2)