What is the key difference between a permutation and a combination?

Order matters in permutations (e.g. 1st, 2nd, and 3rd place finishes, or a passcode). Order does NOT matter in combinations (e.g. picking a hand of 5 cards or selecting a 3-person delegation).

Why is a 'combination lock' mathematically a misnomer?

Because sequence order is critical for unlocking it (e.g. 15-25-35 will not unlock with 25-15-35), a combination lock is mathematically a permutation lock.

What happens when r = n?

When selecting all items (r = n): nPn = n! (the number of complete orderings), while nCn = 1 (there is only one unique way to pick all items together).

Math

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Permutation and Combination Calculator — nPr & nCr Solver

Free permutation and combination calculator. Compute nPr and nCr both with and without replacement, with step-by-step factorial formulas and probabilities.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

The Permutation and Combination Calculator computes the number of ways to choose r items from a total set of n elements, contrasting situations where sequence order matters (Permutations) versus where order does not matter (Combinations).

Total Number of Items (n)

Size of the overall pool / population

Items Selected (r)

Subset size (r ≤ n)

Presets:

What is Permutation and Combination Calculator — nPr & nCr Solver?

How to use it

1️⃣ Enter the total number of items in the set (n).
2️⃣ Enter the number of items to select (r), where r ≤ n.
3️⃣ Instantly view both the Permutations (nPr) and Combinations (nCr) solutions.
4️⃣ Compare results with and without replacement, accompanied by step-by-step algebraic factorial cancellations.

Formula

Without replacement: nPr = n! / (n - r)! and nCr = n! / [r!(n - r)!]. With replacement: Permutations = nʳ; Combinations = (n + r - 1)! / [r!(n - 1)!].

💡 See it in action — a real example

Selecting 2 officers from 6 candidates (n = 6, r = 2): Permutations ₆P₂ = 6! / 4! = 30 ways. Combinations ₆C₂ = 6! / (2! × 4!) = 15 committee pairings.

❓ Common questions

What is the key difference between a permutation and a combination?: Order matters in permutations (e.g. 1st, 2nd, and 3rd place finishes, or a passcode). Order does NOT matter in combinations (e.g. picking a hand of 5 cards or selecting a 3-person delegation).
Why is a 'combination lock' mathematically a misnomer?: Because sequence order is critical for unlocking it (e.g. 15-25-35 will not unlock with 25-15-35), a combination lock is mathematically a permutation lock.
What happens when r = n?: When selecting all items (r = n): nPn = n! (the number of complete orderings), while nCn = 1 (there is only one unique way to pick all items together).