What is the difference between permutations with and without repetition?

Without repetition each item is used once, giving n!/(n-r)!. With repetition every position can be any item, giving n^r. A four-digit PIN allows repeats, so it has 10,000 possibilities rather than 5,040.

Why do permutation counts get so large so fast?

Because the factorial grows faster than any exponential. Ten items arrange 3.6 million ways and twenty arrange 2.4 quintillion, which is why shuffling a deck of 52 almost certainly produces an order no deck has ever been in.

Math & Statisticsprobability

Permutation Calculator

Permutations, with the matching combination count shown so the difference between "order matters" and "order does not" is visible rather than assumed.

Also called: npr calculator, arrangements calculator.

Permutations

720

There are 720 ways to arrange 3 items from 10 when the order matters. Against 120 when it does not.

Combinations, for contrast

120

How this is calculated

Method and background

How this is calculated

The first slot has n choices, the second n−1, and so on for r slots, which is n!/(n−r)!. Combinations divide that by r! because the r chosen items can themselves be arranged r! ways and those arrangements are considered identical.

nPr = n! / (n-r)!

n: Items available
r: Items chosen

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

10P3

Choose from (n): 10
Choose how many (r): 3

Permutations720

arithmetic identity: 10x9x8, and 720/3!

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choosing everything is n!

Choose from (n): 5
Choose how many (r): 5

Permutations120

boundary

Open this example

choosing more than exists is zero

Choose from (n): 3
Choose how many (r): 5

Permutations0

degenerate case

Open this example

Frequently asked questions

What is the difference between permutations with and without repetition?: Without repetition each item is used once, giving n!/(n-r)!. With repetition every position can be any item, giving n^r. A four-digit PIN allows repeats, so it has 10,000 possibilities rather than 5,040.
Why do permutation counts get so large so fast?: Because the factorial grows faster than any exponential. Ten items arrange 3.6 million ways and twenty arrange 2.4 quintillion, which is why shuffling a deck of 52 almost certainly produces an order no deck has ever been in.