Permutation Calculator
Calculate arrangements where order matters: P(n,r)
Enter total items (n)
Input the total number of items in your set. This is the complete pool you're selecting from.
Enter items to select (r)
Input how many items you want to arrange. This must be less than or equal to n.
Click Calculate
The calculator computes P(n,r) — the number of possible arrangements where order matters.
| Scenario | n | r | P(n,r) |
|---|---|---|---|
| Podium finishes (1st, 2nd, 3rd) from 8 runners | 8 | 3 | 336 |
| Arrange 5 books on a shelf | 5 | 5 | 120 |
| Pick president and VP from 10 members | 10 | 2 | 90 |
| 4-digit codes from digits 0-9 | 10 | 4 | 5,040 |
| Seating 6 people at a table | 6 | 6 | 720 |
| Top 3 songs from playlist of 20 | 20 | 3 | 6,840 |
What Is a Permutation?
A permutation is an arrangement of items where order matters. ABC and BAC are different permutations of the same three letters. The key question permutations answer is: "How many different ways can I arrange r items selected from n total items?"
The Permutation Formula
The formula for permutations is P(n,r) = n! / (n-r)!. Factorial (written as !) means multiply all positive integers up to that number. So 5! = 5 × 4 × 3 × 2 × 1 = 120. For P(8,3): 8! / (8-3)! = 8! / 5! = 8 × 7 × 6 = 336.
Permutations vs Combinations
Permutations count arrangements where order matters. Combinations count selections where order doesn't matter. Picking a committee of 3 from 10 people is a combination. Picking president, VP, and treasurer from 10 people is a permutation — the positions are distinct.
Rankings and Positions
Use permutations for race results, contest rankings, or any scenario where 1st, 2nd, 3rd are different outcomes.
Passwords and Codes
Arranging digits or characters where sequence matters. The code 1234 differs from 4321.
Seating Arrangements
Who sits where matters. Different seatings of the same people count as different permutations.
Scheduling
Order of tasks, classes, or appointments. The sequence affects the outcome.
What does P(n,r) mean?
P(n,r) means "permutations of n items taken r at a time." It counts how many ways you can arrange r items selected from n total items when order matters.
When do I use factorial in permutations?
Factorial appears in the permutation formula: P(n,r) = n! / (n-r)!. When r equals n (arranging all items), it simplifies to just n!.
How is permutation different from combination?
Permutations care about order; combinations don't. ABC and BAC are 2 different permutations but 1 combination. Use permutations for arrangements, combinations for selections.
Can r be larger than n?
No. You can't select more items than you have. If r > n, the permutation equals 0. The calculator requires r ≤ n.
What is 0! (zero factorial)?
By definition, 0! = 1. This makes the formulas work correctly. P(n,n) = n! / 0! = n! / 1 = n!, which is correct for arranging all n items.