Permutation Calculator (nPr)
Count permutations nPr with or without repetition, or the distinct arrangements of a word, as exact numbers — and list every arrangement for small cases.
A permutation counts the ways to arrange items when order matters — race finishing positions, PIN codes, seating plans or passwords. Choose arrangements without repetition (nPr), with repetition allowed (nʳ), or the distinct arrangements of a word with repeated letters. Results are exact even when they run to dozens of digits, and small cases can be listed in full.
How to use this tool
- Pick the type: no repetition, repetition allowed, or letters of a word.
- Enter n (items available) and r (positions to fill), or type the word.
- Optionally list the items, such as A, B, C, D, to see every arrangement when there are 360 or fewer.
- Read the count; for nPr the matching combination count nCr is shown too.
Formula
nPr = n! ÷ (n − r)!
With repetition: nʳ · Letters of a word: n! ÷ (n₁! × n₂! × …)
n! (n factorial) is n × (n − 1) × … × 1, with 0! = 1. In the word formula, n₁, n₂ and so on are the counts of each repeated letter; dividing by them removes arrangements that look identical. Combinations ignore order, so nCr = nPr ÷ r!.
Example
Gold, silver and bronze among 10 runners: 10P3 = 10 × 9 × 8 = 720 possible podiums, but only 720 ÷ 3! = 120 possible groups of three medallists. A 4-digit PIN where digits may repeat has 10⁴ = 10,000 options, and the letters of BANANA make 6! ÷ (3! × 2!) = 60 distinct words.
Frequently asked questions
What is the difference between a permutation and a combination?
Order matters in a permutation and not in a combination. Choosing a president, secretary and treasurer from 10 people is 10P3 = 720; choosing a three-person committee is 10C3 = 120.
Why is 0! equal to 1?
There is exactly one way to arrange nothing — the empty arrangement. Defining 0! = 1 also keeps formulas such as nPn = n! ÷ 0! working.
How many ways can a deck of 52 cards be shuffled?
52! ≈ 8.0658 × 10^67. Choose no repetition with n = 52 and r = 52 to see all 68 digits.
When should I allow repetition?
When an item can be used again in another position, as with digits in a PIN or characters in a password. Without repetition each item is used at most once, as with people in seats.
Last updated: 2026-10-30