Permutation & Combination Calculator
nPr, nCr and n! for counting arrangements and selections — the foundation of combinatorics and probability.
Permutations count ordered arrangements (podium finishes, passwords, seatings); combinations count unordered selections (lottery picks, committees, hands). The relationship: nPr = nCr × r!. Factorials grow explosively — 20! already exceeds 2×10¹⁸ — so large n loses precision in floating point.
Formula
About Permutation & Combination Calculator
Permutations and combinations are the two ways to count possibilities, and the difference is simply whether order matters. Permutations (nPr) count ordered arrangements — podium finishes, passwords, seatings — while combinations (nCr) count unordered selections — lottery numbers, committees, card hands. This calculator computes both plus the factorial n!, with the worked relationship nPr = nCr × r!. It's the foundational counting tool for probability, statistics, and any 'how many ways can this happen?' question, built for learning the concepts as much as getting the number.
How to use Permutation & Combination Calculator
- 1Enter your values into Permutation & Combination Calculator — sensible, domain-typical defaults are pre-filled so you see a real result immediately.
- 2The result recomputes live using the formula shown on the page; there is no button to press.
- 3Adjust any input to compare scenarios, then read the worked example to see the substituted numbers.
Why use Permutation & Combination Calculator?
- ✓Computes Permutation & Combination instantly in your browser — no sign-up, no upload, no server round-trip.
- ✓100% free and unlimited, with the exact formula shown: nPr = n! / (n-r)! (order matters).
- ✓Runs entirely client-side, so every value you enter stays private on your device.
- ✓Live recompute as you type, with a worked example and authoritative references for trust.
Frequently asked questions
When do I use permutations vs combinations?+
Ask whether order matters. If arranging ABC differently (ABC vs CBA) counts as distinct, use permutations (nPr): race finishes, passwords, rankings, seating. If only the set chosen matters regardless of order, use combinations (nCr): lottery draws, poker hands, picking a committee. The same selection counts once as a combination but r! times as permutations.
What's the relationship between nPr and nCr?+
nPr = nCr × r!. Permutations are combinations multiplied by the number of ways to order the r chosen items. So permutations are always ≥ combinations, equal only when r is 0 or 1. This is why combinations are smaller — they collapse all the orderings of each selection into a single count.
Why do factorials grow so fast?+
Each term multiplies by the next integer, so growth is super-exponential: 10! is 3.6 million, 20! is 2.4×10¹⁸, 70! exceeds the number of atoms in the observable universe. This explosive growth is why brute-forcing arrangements is hopeless for even modest n, and why combinatorial problems demand clever counting rather than enumeration.
Are these used in probability?+
Constantly — most discrete probability is 'favorable outcomes ÷ total outcomes', and counting each requires permutations or combinations. The chance of a specific lottery combination is 1/nCr; the chance of a particular ordering is 1/nPr. Mastering this counting is the prerequisite for computing real probabilities, which our probability and binomial tools build on.
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