Permutation Calculator (nPr)
Calculate permutations nPr — the number of ordered arrangements of r items from n. Free permutation calculator with and without repetition and step-by-step working.
Updated 2026-06-14 · Free · No sign-up · Runs privately in your browser
Show the formula & steps
How the Permutation Calculator Works
Enter the total number of items n and how many you are arranging r, and this calculator returns the number of permutations (nPr) — ordered arrangements without repetition — plus the count with repetition (nʳ). The working is shown step by step.
The Formula
nPr = n! ÷ (n − r)! = n × (n − 1) × … × (n − r + 1) With repetition: nʳ
In a permutation, order matters: ABC and CBA are counted as two different arrangements.
Worked Example
How many ways can you arrange 3 items from 10 (for example, awarding 1st, 2nd and 3rd place to 10 contestants)?
- nPr = 10! ÷ (10 − 3)! = 10! ÷ 7!
- = 10 × 9 × 8 = 720 ordered arrangements
If positions could repeat, it would be 10³ = 1,000 instead.
Permutation Examples
| n | r | nPr (no repeat) | nʳ (with repeat) |
|---|---|---|---|
| 5 | 2 | 20 | 25 |
| 10 | 3 | 720 | 1,000 |
| 6 | 6 | 720 | 46,656 |
| 52 | 5 | 311,875,200 | 380,204,032 |
Where Permutations Are Used
Permutations count outcomes in any situation where order matters: race finishes, seating orders, passwords and PINs, schedules, and arrangements of distinct objects. They are a building block of probability, where the number of favorable orderings is divided by the total number of orderings to find the chance of an event.
Frequently asked questions
What is a permutation?+
A permutation is an arrangement of items where order matters. Choosing 3 runners for gold, silver and bronze from 10 is a permutation because the order of finish changes the outcome. The number of such arrangements is written nPr.
What is the formula for permutations?+
The number of permutations of r items chosen from n is nPr = n! ÷ (n − r)!. For 10 items taken 3 at a time, that is 10! ÷ 7! = 10 × 9 × 8 = 720 ordered arrangements.
What is the difference between a permutation and a combination?+
Permutations count arrangements where order matters; combinations count selections where order does not. Picking 3 people for specific roles is a permutation, while picking 3 people for an unranked committee is a combination. There are always more permutations than combinations for the same n and r.
What does permutation with repetition mean?+
With repetition, an item can be chosen more than once, like digits in a PIN. The count is then nʳ instead of n! ÷ (n − r)!. A 4-digit PIN using digits 0–9 has 10⁴ = 10,000 permutations because each position can repeat any digit.
Why can't r be larger than n without repetition?+
Without repetition you cannot arrange more items than you have, so r must be at most n. If you need to allow positions to repeat — for example arranging 5 letters from a 3-letter alphabet — use the with-repetition count nʳ instead.