Permutation Calculator
Use this calculator when order matters — rankings, seat assignments, PIN patterns, route sequences, and scheduling arrangements.
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Result
The formula
Standard permutations choose and order r items from n: nPr = n!/(n−r)!. With repetition allowed, each slot has n choices so total arrangements are n^r. Circular permutations fix rotational symmetry, giving (n−1)! distinct seatings around a round table.
Worked example
- Standard permutation with n = 5 and r = 3.
- nPr = 5!/(5−3)! = 5!/2! = (5×4×3×2×1)/(2×1).
- Result: 60 ordered arrangements.
Result: 5P3 = 60
How permutation counts are computed
Permutation formulas count ordered outcomes. The same selected items in a different order count as a different result.
Standard permutation (nPr)
Use this when you pick r distinct items from n distinct options and order matters. Valid only when n and r are non-negative integers with n ≥ r.
Permutation with repetition (n^r)
Use this when each position can reuse any of the n options (e.g., lock codes where digits can repeat). Total arrangements are n multiplied by itself r times.
Circular permutation ((n−1)!)
For round-table seating, rotations are equivalent (ABC is the same circle as BCA), so one position is fixed and arrangements reduce to (n−1)!.
Input constraints
This calculator expects whole numbers. Very large n can exceed JavaScript safe integer precision, so results are intended for practical counting ranges.
Interesting facts
Order changes the answer
Choosing A, B, C is one combination, but ABC, ACB, BAC, BCA, CAB, and CBA are six different permutations.
Factorials grow fast
10! is already 3,628,800. That growth is why permutation counts get large quickly.
Circular cases remove duplicates
In a circle, rotating everyone one seat does not create a new arrangement, so counts are smaller than linear seatings.
Frequently asked questions
Permutation cares about order; combination does not. If ranking or position matters, use permutation.
Use nPr when you select r items from n without repeating items, and arrangement order is important.
Use repetition when an option can appear more than once across positions, such as repeated digits in a code.
Because rotating all people together gives the same seating pattern, one reference position is fixed to avoid overcounting.
No. Permutation counting uses non-negative integers.