Enter n (total objects) and r (objects chosen) to calculate P(n,r) — the number of ordered arrangements possible from your set. Choose between permutations without repetition (P(n,r) = n! / (n−r)!) and permutations with repetition (n^r). Results include the permutation count, the corresponding combination count, and a step-by-step breakdown of the formula.
Permutations P(n,r)
--
Combinations C(n,r)
--
n! (n Factorial)
--
r! (r Factorial)
--
(n−r)! Factorial
--
A permutation is an arrangement of items from a set where the order matters. For example, choosing 2 letters from {A, B, C} gives AB and BA as two different permutations, even though they contain the same elements.
The formula is P(n,r) = n! / (n−r)!, where n is the total number of objects and r is the number chosen. The exclamation mark denotes a factorial, meaning you multiply all integers from 1 up to that number.
When repetition is allowed, the formula simplifies to P(n,r) = n^r. This is because each of the r positions can independently hold any of the n objects.
In a permutation, the order of selection matters — AB and BA are different. In a combination, order does not matter — AB and BA are the same. The combination formula is C(n,r) = n! / (r! × (n−r)!).
When n = r, the formula P(n,r) = n! / (n−r)! simplifies to n! / 0! = n!, since 0! equals 1. This means all n objects are arranged in every possible order.
Using 62 alphanumeric characters (a–z, A–Z, 0–9) with repetition allowed, the number of possible 10-character passwords is 62^10 = 839,299,365,868,340,224 — over 839 quadrillion combinations.
By mathematical convention, 0! = 1. This ensures consistency in factorial-based formulas. For example, when r = n, the denominator (n−r)! = 0! must equal 1 so that P(n,n) = n! correctly counts all arrangements.
No. If r > n and repetition is not allowed, the permutation is undefined (you cannot choose more items than exist in the set). This calculator will return an error if you enter r > n without repetition enabled.