| n factorial, n! | 3,628,800 |
| Permutations, nPr = n! / (n − r)! | 720 |
| Combinations, nCr = n! / (r! (n − r)!) | 120 |
| Permutations with repetition, nʳ | 1,000 |
| Combinations with repetition, C(n + r − 1, r) | 220 |
Results are exact integers, shown in full with thousands separators.
Use a permutation when order matters, such as podium places or passwords built from distinct characters. Use a combination when order does not matter, such as picking a committee or a five-card poker hand. Five-card hands from a 52-card deck are C(52, 5) = 2,598,960; the same five cards arranged in order would be P(52, 5) = 311,875,200.
Count permutations, combinations and factorials exactly. Permutations count ordered selections and combinations count unordered ones. This calculator returns nPr, nCr and n factorial, plus the counts with repetition, using exact integer arithmetic so even large results are not rounded. The definitions follow chapter 26 of the NIST Digital Library of Mathematics Functions, which covers combinatorial analysis.
Most counting mistakes come from mixing up order and repetition. The calculator gives every count for the same n and r, so you can see which one matches your problem. 100% free, no registration, and complete privacy — everything runs locally in your browser, so your data never touches a server.
Permutations and combinations side by side, with the formulas written out.
nʳ and C(n + r − 1, r) for selections where items may repeat.
Results such as C(100, 50) are printed in full, not rounded or in scientific notation.
All counting happens in your browser.
Averages and modes for a list of numbers.
Standard normal probabilities for the large-sample counts in a test.
Deal a list of names into random teams.
Draw random whole numbers from any range.
A permutation counts ordered selections, so ABC and BAC are different. A combination counts unordered selections, so they are the same group. Use nPr when order matters and nCr when it does not.
It is 2,598,960, the number of five-card poker hands from a 52-card deck. The order of the cards in a hand does not change the hand, so a combination is the right count.
By definition, so that formulas stay consistent. C(n, 0) = 1 and P(n, 0) = 1, since there is exactly one way to choose nothing.
Counts grow factorially. P(52, 5) is 311,875,200, and 52! has 68 digits. Exact integer arithmetic avoids the rounding that floating-point numbers would introduce.
This version handles n and r up to 1000. That covers textbook problems and most practical counting questions.
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