Permutation nPr
Ordered arrangements
About This Calculator
A permutation counts the number of ways to arrange r items from a set of n items when order matters. Unlike combinations, swapping the order produces a different permutation. Permutations are used in counting passwords, race finishing orders, seating arrangements, and sequential processes.
Formula
P(n,r) = n! / (n−r)! = nPr
P(n,n) = n! (arrange all n items)
Relation to combinations: P(n,r) = C(n,r) × r!
Example Calculation
How many ways can 3 runners finish in first, second, and third from a group of 8?
- n=8, r=3
- P(8,3) = 8! / (8−3)! = 8! / 5! = 8×7×6 = 336
There are 336 possible finishing orders
Permutation Values P(n,r)
| n \ r | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| 5 | 5 | 20 | 60 | 120 | 120 |
| 6 | 6 | 30 | 120 | 360 | 720 |
| 7 | 7 | 42 | 210 | 840 | 2520 |
| 8 | 8 | 56 | 336 | 1680 | 6720 |
| 10 | 10 | 90 | 720 | 5040 | 30240 |
Frequently Asked Questions
What is the difference between permutation and combination?
In a permutation, order matters: (A,B,C) is different from (C,B,A). In a combination, order doesn't matter: {A,B,C} is the same group regardless of arrangement. P(n,r) = C(n,r) × r! because each combination generates r! permutations.
What is a factorial?
n! (n factorial) is the product of all positive integers from 1 to n. 5! = 5×4×3×2×1 = 120. By convention 0! = 1. Factorials grow extremely fast: 20! ≈ 2.4 × 10¹⁸.
How many ways can n people sit in n seats?
This is a full permutation: P(n,n) = n!. For 4 people in 4 seats: 4! = 24 ways. For 10 people: 10! = 3,628,800. This is why 'shuffling' a deck of 52 cards produces 52! ≈ 8×10⁶⁷ possible orderings.
What are circular permutations?
In circular permutations, rotating all items doesn't create a new arrangement. The number of ways to arrange n people in a circle is (n−1)! instead of n!. For 5 people around a table: (5−1)! = 24 arrangements.