Permutations Calculator (nPr)
How many ordered arrangements of r items from n — nPr — with the combination count and the full-ordering count alongside.
A permutation counts arrangements: first, second and third place are three different outcomes for the same three runners.
How the permutations calculator (npr) works
A permutation counts arrangements: first, second and third place are three different outcomes for the same three runners. The count is n × (n − 1) × … down r terms, which is n! ÷ (n − r)!.
When r equals n you are arranging everything, and the answer is n! — which grows terrifyingly fast. Fifteen items already have more orderings than there are seconds in forty thousand years.
Formula: P(n, r) = n! / (n − r)!
Worked examples
| Inputs | Permutations P(n, r) | Note |
|---|---|---|
| Gold, silver, bronze from 10 runners | 720 | 720 |
| Arrange all 5 items | 120 | 120 |
| A 4-digit PIN from 10 digits, no repeats | 5,040 | 5,040 |
FAQFrequently asked questions
What is nPr?
The number of ways to fill r ordered positions from n distinct items: n! ÷ (n − r)!.
When do I use permutations rather than combinations?
When the order of the result matters — podium places, passwords, seating, rankings.
How many ways can 5 people sit in a row?
5! = 120.
What if repeats are allowed?
Then it is simply nʳ — a 4-digit PIN from 10 digits gives 10,000, not 5,040.
Why does the count grow so fast?
Each new item multiplies every existing arrangement. 10! is 3.6 million; 20! is bigger than the age of the universe in seconds.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.