Numbers
Permutation Calculator
A permutation calculator works out how many ways you can arrange r items chosen from a set of n when the order matters — the quantity written P(n, r) or "n permute r". Enter the totals and it returns the exact count using big-integer arithmetic, so large values stay precise. Students use it to check combinatorics homework, teachers to create examples, and anyone reasoning about rankings, passwords, race finishes, or seating to count ordered possibilities. Permutations apply whenever sequence is significant — first, second, and third place differ from third, second, and first — which makes them larger than the matching combination count. The result spells out that permutations count ordered arrangements, the key distinction from combinations. Use it to verify a calculation, explore how many orderings are possible, or see why order multiplies the possibilities so quickly.
How to use
- Choose your options above
- Click Generate
- Copy your result
Detailed instructions
- Enter the total number of items, n.
- Enter how many you are arranging, r.
- Click Generate to compute P(n, r) exactly.
- Use the result to check work or count orderings.
Use Cases
- •Checking combinatorics homework on arrangements
- •Counting possible rankings or race finishes
- •Computing password or code possibilities
- •Generating worked examples for a class
- •Counting ordered seatings or schedules
Tips
- →Use permutations when order matters, combinations when it does not.
- →P(n, r) equals C(n, r) × r!.
- →Pair with the factorial tool to see the formula behind it.
- →Large counts stay exact thanks to big-integer maths.
FAQ
what is a permutation
A permutation is an ordered arrangement. P(n, r) counts how many ways you can arrange r items chosen from n when sequence matters — so first, second, third is different from third, second, first.
how is it different from a combination
Permutations count order, combinations ignore it. For the same n and r, there are always at least as many permutations as combinations — exactly r! times as many, since each chosen group can be ordered in r! ways.
are large results exact
Yes. The tool uses big-integer arithmetic, so even large permutation counts are computed exactly to the final digit rather than rounded to a floating-point approximation.
What is the formula for permutations?
The number of ways to arrange r items from n, where order matters, is P(n, r) = n! / (n − r)! — so P(10, 3) = 10 × 9 × 8 = 720. The calculator applies this formula and shows the result with the working, so you can both get the answer and see how it is derived rather than just trusting a black-box number.
When would I use permutations in real life?
Whenever order matters — the number of possible race finishes (who comes 1st, 2nd, 3rd), seating arrangements, passwords or PINs by position, or ranked shortlists. If rearranging the same items counts as a different outcome, you want a permutation, not a combination. The generator computes P(n, r) for these cases, so you can size a problem's possibilities quickly and correctly.
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