Permutations and Combinations Cheat Sheet

Permutations and Combinations Cheat Sheet

Quick Revision

Permutations count arrangements, where order matters; combinations count selections, where order does not matter. Both are built from the factorial.

1. The Factorial

n! = n × (n − 1) × (n − 2) × … × 2 × 1
0! = 1

2. Permutations (Order Matters)

The number of ways to arrange r items out of n:

ⁿPr = n! / (n − r)!

3. Combinations (Order Does Not Matter)

The number of ways to choose r items out of n:

ⁿCr = n! / [r! × (n − r)!]

4. Key Identities

ⁿCr = ⁿC₍ₙ₋ᵣ₎
ⁿPr = ⁿCr × r!
ⁿC₀ = ⁿCₙ = 1

5. Worked Example

⁵P₂ = 5! / 3! = 20            (arrange 2 of 5, order matters)
⁵C₂ = 5! / (2! × 3!) = 10     (choose 2 of 5, order ignored)

Arrangements of the letters in APPLE (P repeats twice): 5! / 2! = 60.

6. Exam Quick Tips

• Ask "does order matter?" — a line or ranking uses ⁿPr; a team or group uses ⁿCr.
• Divide by k! for every letter that repeats k times when arranging a word.
• ⁿCr is largest when r is close to n/2.

Keep revising with the rest of the Maths section:

• All Maths topics
• Probability Cheat Sheet
• Statistics Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.