Permutation Calculator
Enter n, the number of items, and r, how many you arrange, to count the ordered arrangements nPr. Switch on repetition for cases such as PIN codes.
Last updated November 3, 2026
Result
P(10, 3) = 720
| Permutations (order matters) | 720 |
|---|---|
| Combinations for comparison | 120 |
| n! | 3,628,800 |
| Digits in the answer | 3 |
Show the work
Formula: P(n, r) = n! / (n - r)!
- n = 10, r = 3, repetition not allowed.
- n! / (n - r)! is the product of the r largest factors: 10 x 9 x 8.
- P(10, 3) = 720.
Result: P(10, 3) = 720
How permutations are counted
P(n, r) = n! ÷ (n − r)!
A permutation is an arrangement in which order matters. Choosing Ann then Ben for first and second place is a different permutation from Ben then Ann. To count them, multiply the number of choices for each position: n for the first, n − 1 for the second, and so on for r positions.
That product equals n! ÷ (n − r)!, where n! means n × (n − 1) × … × 1. For 10 items taken 3 at a time it is 10 × 9 × 8 = 720. When r equals n, every item is used and the answer is simply n!, so 5 items can be arranged in 120 ways.
With repetition allowed, each position can use any of the n items again, so the count is n^r. A 4-digit PIN with 10 digits to pick from has 10^4 = 10,000 possibilities. The calculator returns exact whole numbers, even for very large results.
See how we calculate for the formulas, sources and review schedule behind every calculator.
Also useful alongside this calculator: Probability Calculator, Statistics Calculator and Sample Size Calculator.
When order matters
Use permutations for rankings, seating plans, passwords and race results, where each position means something different. Eight runners competing for gold, silver and bronze can finish in 8 × 7 × 6 = 336 ways.
If only the group matters and not the order, use combinations instead. The two are linked by P(n, r) = C(n, r) × r!, so 720 permutations of 3 from 10 correspond to 120 groups of 3, each arranged in 6 ways.
With and without repetition
Without repetition each item is used at most once, so r cannot be larger than n. Three-letter codes from 26 letters with no letter reused number 26 × 25 × 24 = 15,600.
With repetition a letter can appear again, so there are 26 × 26 × 26 = 17,576 codes. With repetition r may exceed n, for example 3 symbols in strings of length 4 give 81 arrangements.
Very large results
Counts grow quickly. The exact value of 20! is 2,432,902,008,176,640,000, which is more than a normal floating-point number can store without rounding. The calculator uses exact integer arithmetic, so no digit is lost.
Results of more than 24 digits appear in scientific notation, with the digit count in the table. The number of ways to order a deck of 52 cards is about 8.06581 × 10^67. The limit is n and r of 1,000 each.
Arranging items that repeat
If some items are identical, divide by the factorial of each repeat. The letters of LEVEL include two Ls and two Es, so there are 5! ÷ (2! × 2!) = 120 ÷ 4 = 30 distinct arrangements. This page counts distinct items; do the division yourself for repeated letters.
How to use the Permutation Calculator
- Enter n, the total number of distinct items.
- Enter r, how many of them you arrange, and set whether items can repeat.
- Read nPr, with the combination count shown for comparison.
- Open Show the work for the multiplication, then use Copy, Print or Share.
Worked example
Ten contestants compete for gold, silver and bronze. There are 10 choices for gold, 9 for silver and 8 for bronze, so P(10, 3) = 10 × 9 × 8 = 720 podiums. The unordered groups of three number 120, and 720 ÷ 6 = 120.
| n | r | Repetition | Arrangements |
|---|---|---|---|
| 5 | 5 | No | 120 |
| 6 | 2 | No | 30 |
| 8 | 3 | No | 336 |
| 10 | 3 | No | 720 |
| 10 | 4 | Yes | 10,000 |
Frequently asked questions
What is the difference between a permutation and a combination?
In a permutation the order of the chosen items matters, in a combination it does not. Permutations are always at least as many as combinations, by a factor of r!.
What does nPr mean?
nPr, also written P(n, r), is the number of ordered arrangements of r items chosen from n distinct items, equal to n! ÷ (n − r)!.
Why does it say r cannot be larger than n?
Without repetition you cannot pick more items than exist. Choose Yes for repetition if items can be reused, which allows any r.
What is 0! and what is P(n, 0)?
0! is 1 by definition, and P(n, 0) is 1 because there is exactly one way to arrange nothing, the empty arrangement.
Are my numbers stored or sent anywhere?
No. The calculation runs in your browser and nothing you enter is saved or uploaded unless you copy or share a link yourself.