What is the difference between a permutation and a combination?

Permutation (nPr): order matters. Choosing 3 people for President, VP, Secretary from 10 — each arrangement is different. Formula: nPr = n! / (n−r)!. Combination (nCr): order doesn't matter. Choosing 3 people for a committee from 10 — the same group of 3 people is counted once regardless of order. Formula: nCr = n! / (r! × (n−r)!). The relationship: nPr = nCr × r!

How is the notation different in UK, US, and IB?

US: C(n,r) or ₙCᵣ for combinations; P(n,r) or ₙPᵣ for permutations. UK A-Level: ⁿCᵣ (superscript n, subscript r) — appears on Edexcel and AQA formula sheets. IB: P(n,r) and C(n,r) or nCr notation. The binomial coefficient is also written as (n choose r) or as the binomial symbol with n on top and r below — these are all the same as nCr.

🇺🇸 US: C(n,r) / P(n,r) 🇬🇧 UK: ⁿCᵣ / ⁿPᵣ 🌐 IB: nCr notation

Permutation & Combination Calculator

Calculate nPr (permutations — order matters) and nCr (combinations — order doesn't matter) with step-by-step working. Exact results using BigInt for any n up to 170.

Quick Reference

nPr = n!/(n−r)! · nCr = n!/(r!(n−r)!) · nPr = nCr × r! · Order matters → permutation. Order irrelevant → combination.

nPr (permutations) = order matters · nCr (combinations) = order doesn't matter
UK notation: ⁿPᵣ and ⁿCᵣ · US notation: P(n,r) and C(n,r) · IB: P_rⁿ and C_rⁿ

n (total items)

r (chosen items)

Country / Curriculum	Permutation Notation	Combination Notation	Binomial Symbol
🇺🇸 US (AP Stats, SAT)	P(n,r) or nPr	C(n,r) or nCr	(n choose r)
🇬🇧 UK A-Level (Edexcel/AQA)	ⁿPᵣ	ⁿCᵣ	(n r) — binomial coefficient
🌐 IB Mathematics	P(n,r)	C(n,r) or nCr	Same as nCr
🇩🇪 Germany (Abitur)	n! / (n-k)! — Variation	Kombination ₙCₖ	(n k)
🇯🇵 Japan (数学B)	nPr — 順列	nCr — 組合せ	(n r)
🇫🇷 France (Terminale)	Arrangement Aⁿₖ	Combinaison Cⁿₖ	(n k)

Frequently Asked Questions

How do I know whether to use permutations or combinations?

Ask: "Does the order matter?" If yes → permutation. If no → combination. Examples: Passwords (order matters) → permutation. Lottery numbers (which numbers you pick, not order) → combination. Ranking candidates (1st, 2nd, 3rd) → permutation. Choosing a committee → combination. Seating arrangements → permutation. Choosing pizza toppings → combination.

Why does the UK use ⁿCᵣ notation but the US uses C(n,r)?

Both notations mean exactly the same thing — it's purely a typographic convention adopted by different education systems. UK Edexcel and AQA exam boards use ⁿCᵣ (superscript-subscript format) on their formula sheets. US textbooks and AP Statistics use C(n,r) or the horizontal notation nCr. IB Mathematics uses both interchangeably. In computing (Python, Excel), it's written as comb(n, r) or COMBIN(n, r).

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APA

WorldCalculators.org. (n.d.). Permutation & Combination Calculator: nPr & nCr with Steps. https://worldcalculators.org/calculators/permutation-combination/

MLA

“Permutation & Combination Calculator: nPr & nCr with Steps.” WorldCalculators.org, worldcalculators.org/calculators/permutation-combination/.

BibTeX

@misc{worldcalculatorscalculatorspermutationco,
  title        = {Permutation & Combination Calculator: nPr & nCr with Steps},
  author       = {{WorldCalculators.org}},
  year         = {n.d.},
  howpublished = {\url{https://worldcalculators.org/calculators/permutation-combination/}}
}

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