Example 1
- n = 5
120
How many permutations of 5 items? P(5) = 120.
Permutation calculator
Type n. A permutation is an arrangement of n distinct items in a line. P(n) = n!. P(5) = 120. P(4) = 24. P(0) = 1, because the empty list has one way: arrange nothing.
The calculator uses the integer part of n. A negative n returns 0. This is not a combination and not a variation with k smaller than n.
Results
Enter data and click Calculate.
How it works
A permutation of n items is one way to line all of them up. Five different books on a shelf have 120 orders, because P(5) = 5! = 120. Four people in a queue have 24 line-ups. The symbols P(n) and n! are the same number.
The formula is the factorial: you multiply 1·2·…·n. 0! = 1, so P(0) = 1. An empty list has nobody to shuffle, and the convention leaves one empty arrangement. 1! = 1: one person, one queue.
The field is n. The calculator uses the integer part: 5.7 counts as 5 and returns 120. A negative n in this implementation returns 0, not 1. On the factorial card a minus drops to 0! = 1. Different branch, different result on a minus.
A combination C(n,k) picks a subset with no order. A variation V(n,k) lines up only k of n. A permutation takes everyone: that is V(n,n). So P(4) = V(4,4) = 24.
The header units switch does not convert anything here. 5 stays 120 whatever the label says. This is a count of queues, not a length.
P(6) = 720. P(3) = 6. P(2) = 2. P(7) = 5040. P(10) = 3628800. P(0) and P(1) both return 1, for two different reasons.
Formula
P(n) = n! = 1 · 2 · … · n, and P(0) = 1
How to use
- Type n, for example 5.
- Click Calculate. P(5) = 120.
- P(4) = 24. P(0) = 1.
- A negative n returns 0 in this calculator, not 0!.
- Choosing k of n with no order lives on combinations. Here you arrange everyone.
P(5) = 120
A permutation P(n) = n!. P(5) = 120. P(4) = 24. P(0) = 1. Arrangements of all n items, not a subset.
- permutation
- One line-up of n distinct things. P(5) = 120. Order makes a new result.
- P(n)
- The same number as n!. P(4) = 24. P(n) = V(n,n). A negative n returns 0 here.
- n
- The number of items. The calculator uses the integer part. 5.2 counts as P(5) = 120.
Examples
Example 2
- n = 4
24
How many queues of 4 people? P(4) = 24.
Example 3
- n = 0
1
Is P(0) zero? No. The empty list has one arrangement.
Related calculators
Common questions
What is P(5)?
120. Five distinct things have 120 orders. That is the same 5! as on the factorial card.
What does permutation mean?
An arrangement of all n distinct items. Order makes a new result. Anna-Bartek is not Bartek-Anna.
What are P(4) and P(0)?
P(4) = 24. P(0) = 1. The empty list has one way: arrange nothing.
How is P(n) different from V(n,k)?
A permutation lines everyone up, so P(n) = V(n,n). A variation can take only k seats from n.
Does a negative n give 1?
Not in this calculator. A negative n returns 0. On the factorial card a minus drops to 0! = 1.
Does 5.7 give a different permutation from 5?
No. The calculator uses the integer part, so 5.7 counts as P(5) = 120.
Why is 0! on an empty list?
So the n! formula still holds at zero. One empty arrangement, not “no arrangements”.
What are P(6) and P(3)?
P(6) = 720. P(3) = 6. P(2) = 2. Each next n multiplies by n.
Is this combinations?
No. A combination ignores order. A permutation keeps order and takes the whole set.
What is P(5) good for in a queue?
Five different people can stand in 120 orders. Two of five with no order is C(5,2) = 10, on combinations.
Knowledge sources
The calculator computes the same formula as the definition below.
Page updated in 2026.