THE NUMORIX GUIDE
How to use the Permutation Calculator
Last reviewed September 14, 2026
What this calculator does
For n distinct items choosing and ordering r of them, P(n,r) = n!/(n-r)!.
Formula and method
For n distinct items choosing and ordering r of them, P(n,r) = n!/(n-r)!. The engine also reports C(n,r) = P(n,r)/r!, using exact BigInt arithmetic before formatting.
Variables and inputs
n is the total item count and r is the ordered selection count. The UI requires nonnegative integers with r <= n; n defaults to 10 and r to 3.
Worked example
For n=10 and r=3: P(10,3) = 10! / 7! = 10*9*8 = 720. The corresponding unordered count is C(10,3) = 720/3! = 120.
How to interpret the result
Permutations distinguish order, while combinations treat the same selected group as one outcome.
Common mistakes to avoid
Do not use combinations when seat order or sequence matters. Make sure r is not larger than n and that repeated items are not being treated as distinct without justification.
Assumptions and limitations
The engine caps n at 170 for practical calculation and returns strings when results exceed Number.MAX_SAFE_INTEGER. It assumes all items are distinct.
Practical use and checks
Use permutations when selecting and arranging distinct items, such as assigning people to podium places, filling ordered slots, or creating a no-repeat sequence. Enter n = 10 and r = 3 as a check; the result should be 720 because 10 x 9 x 8 counts the first, second, and third positions. The corresponding combination count is 120, which is smaller because it ignores the six possible orders of each three-item group. Confirm that r is no larger than n before interpreting the result. The output counts arrangements, not probabilities by itself. To turn it into a probability, define the sample space and favorable arrangements separately, and do not assume every arrangement is equally likely unless the process supports that assumption. The engine treats items as distinct and uses integer inputs; repeated letters, identical products, or restrictions such as two people not sitting together require a different count. Results grow rapidly, so a rounded display can conceal a very large exact value, even though the implementation uses BigInt in its core calculation for many cases. The n = 0, r = 0 case has one empty arrangement mathematically, while negative or fractional input is outside the intended interface.