Permutation Calculator
When Order Matters
A permutation counts how many distinct ordered arrangements are possible when selecting r items from a group of n — first place, second place, and third place in a race are three different permutations even if the same three runners are involved. This calculator computes nPr directly from the factorial-based formula, which is far faster than listing every arrangement by hand once n and r grow past single digits.
The Formula
Both n and r must be non-negative whole numbers, and r cannot exceed n — you can't select more items than exist in the group. Dividing by (n−r)! rather than including it as a separate factor cancels out all the arrangements of the items that weren't selected.
Where Permutations Apply
- Race and competition results — counting the number of possible finishing orders for the top positions.
- Password and code generation — counting arrangements where digit or character order matters and repetition isn't allowed.
- Scheduling — the number of ways to sequence a subset of tasks or appointments.
- Seating arrangements — assigning specific people to specific ordered seats.
| n | r | nPr |
|---|---|---|
| 10 | 3 | 720 |
| 5 | 5 | 120 |
When r = n, nPr equals n! exactly, since every item is being arranged with none left over.
How to Use This Calculator
- Enter the Total Items (n).
- Enter the Items Chosen (r).
- Select Calculate to see the number of ordered arrangements.
Related Calculations
When the order of the selection doesn't matter, use the Combination Calculator instead. Both formulas are built from the Factorial Calculator.