About the Permutation Calculator
What is the Permutation Calculator?
A permutation is an arrangement in which order matters. Choosing a president, secretary and treasurer from 10 people, or the first three finishers in a race, are permutations: A-B-C is different from C-B-A.
Enter the number of items available (n) and the number being arranged (r). Without repetition each item can be used once; with repetition items can be reused, as with digits in a PIN. Very large results are shown in scientific notation.
How to use the Permutation Calculator
- Total Items (n) (sample: 10)
- Items Arranged (r) (sample: 3)
- Repetition — allow repetition when an item can be used more than once, as with PIN digits. (sample: Without repetition)
- Press Calculate Permutations (results also refresh as you type). The result card shows the Number of Permutations (nPr), along with Formula used and Order matters.
Formula used
nPr = n! ÷ (n − r)!
- n = the number of items to choose from
- r = the number of positions to fill
- With repetition = nʳ
n! (n factorial) = n × (n − 1) × … × 1, and 0! = 1. nPr is the product of the r largest factors of n!: 10P3 = 10 × 9 × 8 = 720.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Total Items (n) | 10 |
| Items Arranged (r) | 3 |
| Repetition | Without repetition |
| Result | Value |
|---|---|
| Number of Permutations (nPr) | 720 |
| Formula used | 10! / 7! |
| Order matters | Yes |
Tips for using the Permutation Calculator
- If order does not matter (a team, a hand of cards), use the Combination Calculator instead.
- A 4-digit PIN has 10⁴ = 10,000 permutations with repetition, but only 10P4 = 5,040 if no digit repeats.
- Arranging all n items gives n! permutations: 5 books can be ordered on a shelf in 120 ways.