Permutation Calculator
Permutation Calculator
Use permutations when the order of selected items changes the outcome.
Enter your values
Enter values and select Calculate.
Permutation Calculator: Calculate nPr
Use this free Permutation Calculator to find the number of ways to arrange a selected number of items when order matters. Enter the total number of distinct items (n) and the number of positions or items selected (r). The calculator uses the permutation formula nPr = n! ÷ (n − r)!.
How to Use the Permutation Calculator
- Enter the total number of available items in Total items (n).
- Enter how many items or positions you want to select in Items chosen (r).
- Make sure both values are non-negative whole numbers and r is not greater than n.
- Select Calculate to find the number of possible ordered arrangements.
- Select Clear to start a new calculation.
Permutation Formula (nPr)
Here, n is the total number of distinct items, r is the number of items arranged, and ! represents factorial. The formula counts ordered selections without replacement.
Because order matters, arranging A, B and C is different from arranging B, A and C. Each different order is counted as a separate permutation.
Permutation Examples
5P3 = 5! / (5 − 3)! = 5! / 2! = 5 × 4 × 3 = 60 arrangements.
4P2 = 4! / 2! = 4 × 3 = 12 arrangements.
4P4 = 4! / 0! = 24. There are 24 ways to arrange four distinct items.
6P1 = 6! / 5! = 6 possible arrangements.
When Should You Use Permutations?
Use permutations when the order or position of selected items changes the outcome. Examples include assigning first, second and third place in a competition, creating ordered codes from distinct symbols, assigning people to specific roles, and arranging books on a shelf.
Permutation vs Combination
A permutation counts arrangements where order matters. A combination counts selections where order does not matter. For example, selecting three people for a team is usually a combination; assigning those three people to president, secretary and treasurer is a permutation.
Important Rules and Assumptions
- n and r must be whole numbers greater than or equal to zero.
- r must be less than or equal to n.
- This standard nPr formula assumes distinct items and selection without replacement.
- When r = 0, nP0 = 1: there is one way to select and arrange no items.
- When r = n, nPn = n!: all items are arranged.
Frequently Asked Questions
1. What is a permutation?
A permutation is an arrangement of items in which the order matters. Changing the order creates a different permutation.
2. What does nPr mean?
nPr means the number of ordered ways to select and arrange r items from n distinct items without replacement.
3. What is the permutation formula?
The formula is nPr = n! / (n − r)!, where n is the total number of items and r is the number arranged.
4. What is the difference between permutation and combination?
Permutation considers order; combination does not. A ranking is a permutation, while selecting a group is generally a combination.
5. Can r be greater than n?
No. In the standard arrangement without replacement, you cannot select more distinct items than are available. Therefore r must be less than or equal to n.
6. What is nP0?
For any valid non-negative n, nP0 equals 1. There is exactly one way to make an empty selection.
7. What is nPn?
When all n items are arranged, nPn = n!. For example, 4 distinct items can be arranged in 4! = 24 ways.
8. Does a permutation allow repeated items?
This calculator uses the standard nPr formula for distinct items selected without replacement. Problems that allow repetition use a different counting rule.
9. Where are permutations used in real life?
Permutations are useful for rankings, seating arrangements, ordered passwords or codes, assigning distinct roles, and scheduling sequences.
10. How do I calculate permutations by hand?
Multiply n by the next r descending whole numbers. For example, 6P3 = 6 × 5 × 4 = 120.