Permutation and Combination Calculator (nPr, nCr)
Count the ways to choose or arrange items — combinations, permutations, and factorials.
A permutation counts arrangements where order matters; a combination counts selections where it doesn't. From 5 items choosing 2, there are 20 permutations (nPr) but only 10 combinations (nCr), because each unordered pair can be arranged two ways. A factorial, written n!, multiplies every whole number down to 1, so 5! = 120. Enter n and r to get all three.
Results are estimates provided for general information.
Whole numbers only. Permutations (nPr) count ordered picks; combinations (nCr) count unordered ones.
Factorial (10!)
3,628,800
Permutations (10P3)
720
Combinations (10C3)
120
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About the Permutation & Combination Calculator
Permutations and combinations both count the number of ways to pick items from a group, and the difference between them comes down to one question: does the order matter? A permutation counts arrangements where it does — first, second, and third place in a race are a permutation, because swapping two runners gives a different outcome. The formula is nPr = n! / (n − r)!. A combination counts selections where order doesn't matter — a hand of cards is a combination, because the same five cards are the same hand however you sort them. Its formula is nCr = n! / (r! (n − r)!), always a smaller number than the matching permutation. Underpinning both is the factorial, n!, which multiplies every whole number from n down to 1; it's the count of ways to arrange n distinct things in a row. This calculator takes an n (the total number of items) and an r (how many you choose) and returns the permutations, combinations, and n factorial together, so you can see all three at a glance. One honest note on very large inputs: factorials grow astronomically, and beyond about fifteen or sixteen significant digits the result is shown in scientific notation as a close approximation rather than an exact integer — for everyday values like lottery odds or a hand of cards, the answers are exact.
Frequently asked questions
What is the difference between a permutation and a combination?+
A permutation counts arrangements where order matters; a combination counts selections where it doesn't. Choosing 2 from 5 gives 20 permutations but only 10 combinations, because each unordered pair (a combination) can be ordered in 2 ways (its permutations).
What is nCr?+
nCr is the number of combinations — ways to choose r items from n when order doesn't matter — calculated as n! / (r!(n − r)!). It's how you count things like a 5-card poker hand from a 52-card deck (2,598,960 of them).
What is nPr?+
nPr is the number of permutations — ways to arrange r items chosen from n when order matters — calculated as n! / (n − r)!. Choosing and ordering 2 of 5 gives nPr = 20.
What is a factorial?+
A factorial, written n!, is the product of every whole number from n down to 1: 5! = 5 × 4 × 3 × 2 × 1 = 120. It counts the ways to arrange n distinct items in order, and it's the building block of the permutation and combination formulas.
Are the results exact for large numbers?+
For ordinary inputs, yes — exact whole numbers. But factorials grow extremely fast, so past roughly 15–16 significant digits the result is shown in scientific notation as a very close approximation rather than an exact integer. The calculator makes this clear so you're never misled.
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