Permutation Calculator
Find the number of ways to arrange items where order matters. Calculate nPr (permutations) for any combination of n total items and r items to arrange.
Result
- n (Total Items)
- -
- r (Items to Choose)
- -
- Permutations (nPr)
- 0
Formula & Guide
Formula
P
Permutation Formula
nPr = n! / (n-r)!
Factorial of n divided by factorial of (n-r)
=
Alternative Form
nPr = n × (n-1) × ... × (n-r+1)
Product of r descending integers
Formula Variables
n
Total Items
The total number of items available to choose from
r
Items to Arrange
The number of items to arrange (must be ≤ n)
nPr
Permutations
The number of ways to arrange r items from n total items where order matters
!
Factorial
The product of all positive integers up to that number (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120)
Step-by-Step Scenario
Example Scenario
n (Total Items)
5
r (Items to Arrange)
3
1
Understand the Problem
- We want to arrange 3 items from 5 total items, where order matters.
For example, arranging 3 people from 5 candidates for positions 1st, 2nd, 3rd
2
Apply the Formula
- 5P3 = 5! / (5-3)!
- 5P3 = 5! / 2!
- 5P3 = 120 / 2
Additional Examples
Password Arrangement
n: 10 digits
r: 4 positions
Permutations
10P4 = 5,040
Meaning
5,040 different 4-digit codes
Ranking Top 3
n: 8 contestants
r: 3 positions
Permutations
8P3 = 336
Meaning
336 ways to rank top 3
Characteristics of Permutations
Order Matters
Permutations count arrangements where order is important. ABC is different from CBA, so both are counted separately.
More Than Combinations
For the same n and r, permutations always give more arrangements than combinations because order matters in permutations.
Practical Applications
Used in password generation, ranking systems, tournament brackets, seating arrangements, and any scenario where sequence matters.
Factorial Relationship
When r = n, permutations equal n! (factorial of n), representing all possible arrangements of all items.
Important Notes
- Order matters in permutations: ABC is different from CBA, so both are counted as separate permutations.
- r cannot be greater than n. You cannot arrange more items than you have available.
- When r = n, nPr = n! (n factorial), which is the number of ways to arrange all n items.
- Permutations grow very quickly. For example, 10P5 = 30,240, and 20P10 has over 6.7 billion arrangements.
- Use permutations when the sequence or order of items is important, such as rankings, passwords, or positions.
Frequently Asked Questions
Find answers to common questions about permutation calculations.