Permutation Calculator: nPr with and without Repetition
Count ordered arrangements with nPr = n! / (n − r)! and with repetition nʳ, exactly for n up to 1000, with the formula and numbers shown, free in your browser.
Permutations (nPr)
Order matters, each item used at most once.
720
10P3 = 10! / (10 − 3)! = 10 × 9 × 8
Combinations (nCr)
Order does not matter, each item used at most once.
120
10C3 = 10! / (3! × (10 − 3)!)
Permutations with repetition (nʳ)
Order matters, items may repeat.
1,000
nʳ = 10^3
Combinations with repetition
Order does not matter, items may repeat.
220
C(n + r − 1, r) = C(12, 3)
Permutation Calculator
This page opens with permutations first. Enter n, the number of items, and r, how many you arrange, and the calculator gives nPr, the number of ordered arrangements without repetition, as an exact whole number, with the product it comes from. Below it you get nʳ for arrangements with repetition and the matching combination counts. The page starts with n = 10 and r = 3, which gives 720.
The permutation formula
There are n choices for the first position, n − 1 for the second, and so on for r positions. Multiplying them gives nPr. For 10P3 that is 10 × 9 × 8 = 720.
How to use it
Steps
- Type n, the number of distinct items, from 0 to 1000.
- Type r, the number of positions to fill, from 0 to n.
- Read nPr at the top. The formula line shows the factors, with "…" when there are many.
Worked examples
Race podium
Gold, silver and bronze from 10 runners: 10P3 = 10 × 9 × 8 = 720 different podiums.
Arranging books
All 5 books on a shelf: 5P5 = 5! = 120 orders. Choosing and ordering 2 of them: 5P2 = 5 × 4 = 20.
PIN codes, with repetition
A 4-digit PIN can repeat digits, so each position has 10 choices: 10⁴ = 10,000. That is the permutations-with-repetition card, nʳ.
Permutations and combinations
A permutation counts orders, a combination only selections. Every selection of r items can be put in r! orders, so nPr = nCr × r!. For 10 and 3, 720 = 120 × 6. Results are exact big integers, so 1000P1000, which is 1000!, shows all 2,568 digits with a Copy button.
Common counting questions
Permutations answer questions such as how many ways a club of 12 can elect a president, a secretary and a treasurer (12P3 = 12 × 11 × 10 = 1,320), how many 3-letter codes can be made from 26 letters without reusing one (26P3 = 15,600), or how many orders 8 runners can finish in (8P8 = 8! = 40,320). If the roles were the same, for example three equal committee seats, the order would not matter and you would want the combination count instead. If letters or digits may repeat, as in most passwords and PINs, use the nʳ card.
Factorials
The exclamation mark means factorial: n! = n × (n − 1) × … × 2 × 1, with 0! = 1. Factorials grow very fast. 10! is 3,628,800, 20! already has 19 digits, and 170! is the largest that an ordinary calculator can show. This page keeps every digit exactly, up to 1000!.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device.
Related tools
The full Permutation and Combination Calculator shows the same counts, and the Combination Calculator puts nCr first.
Frequently asked questions
What is a permutation?
An ordered arrangement of items. Choosing a gold, silver and bronze winner from 10 runners is a permutation, because the order matters: there are 10 × 9 × 8 = 720 ways.
What is the permutation formula?
nPr = n! / (n − r)!, which is the product of r numbers counting down from n. For 10P3 that is 10 × 9 × 8 = 720.
What is a permutation with repetition?
When items can be used again, each of the r positions has n choices, so there are nʳ arrangements. A 4-digit PIN from 10 digits has 10⁴ = 10,000 possibilities.
What is nPr when r = 0 or r = n?
nP0 = 1, because there is one way to arrange nothing. nPn = n!, every ordering of all the items, so 5P5 = 120.
How do I count arrangements of letters with repeats?
Divide n! by the factorial of each letter's count. BANANA has 6! / (3! × 2!) = 60 arrangements. This page counts arrangements of distinct items; use the formula above for repeated letters.