Permutation Calculator
Count arrangements when order matters, such as rankings, seating, PINs and anagrams. Choose standard, repeated, circular or word arrangements, and see the working and the actual list for small cases.
Calculate
Arrange r of n different items where order matters, with no repeats. nPr = n! ÷ (n − r)!
Result
The formula
Standard permutations choose and order r items from n: nPr = n! ÷ (n − r)!. With repetition, each of r positions has n choices, so totals are n^r. Circular arrangements around a table drop rotations, giving (n − 1)!. Distinct word permutations divide n! by the factorial of each repeated letter’s count.
Worked example
- Standard permutation with n = 5 and r = 3.
- P(5, 3) = 5 × 4 × 3 = 5! ÷ (5 − 3)! = 120 ÷ 2.
- 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 for non-negative integers with n ≥ r.
Permutation with repetition (n^r)
Use this when each position can reuse any of the n options (for example 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 matches BCA), so one position is fixed and arrangements reduce to (n − 1)!.
Word / letter permutations
For a word or multiset of letters, distinct arrangements are n! divided by the product of factorials of each letter’s frequency (so repeated letters do not create duplicate spellings).
Input constraints
Numeric modes expect whole numbers. Word mode accepts letters A–Z only (case-insensitive). Very large results may show in scientific notation.
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.
It divides n! by each repeated letter’s frequency factorial. BOOK has 4! ÷ 2! = 12 distinct spellings because O appears twice.
No. Numeric permutation counting uses non-negative integers.
References
- Permutations — MathWorld Ordered arrangements P(n, k).
- Counting and probability Introductory permutations versus combinations.