Permutation Calculator

Calculate permutations (nPr) where the order of selection matters. Find the number of ways to arrange r items from a set of n items.

Quick Facts

Formula
P(n,r) = n!/(n-r)!
Order matters
Example: P(5,3)
60
5!/(5-3)! = 120/2
vs Combination
Permutation >= Combination
P(n,r) = C(n,r) x r!
Constraint
n >= r >= 0
Both must be non-negative integers

Your Results

Calculated
Permutation P(n,r)
0
Number of ordered arrangements
n! (n factorial)
0
Total items factorial
(n-r)!
0
Remaining items factorial
P(n,r) = n! / (n-r)!

Key Takeaways

  • Permutations count arrangements where order matters
  • Formula: P(n,r) = n! / (n-r)! where n is total items and r is items selected
  • P(n,n) = n! (arranging all items)
  • Permutations are always greater than or equal to combinations: P(n,r) = C(n,r) x r!
  • Use permutations for rankings, sequences, passwords, and ordered selections

What Is a Permutation?

A permutation is an arrangement of objects in a specific order. When calculating permutations, we're counting how many different ways we can arrange a subset of items where the order in which we select them matters.

For example, if you have 5 runners in a race and want to know how many different ways the top 3 can finish, you'd use permutations because first place, second place, and third place are distinct outcomes - ABC is different from CBA.

Real-World Example: Race Results

Total Runners (n) 5
Podium Spots (r) 3
P(5,3) 60

There are 60 different ways the top 3 positions can be filled by 5 runners!

The Permutation Formula Explained

P(n,r) = n! / (n-r)!
P(n,r) = Number of permutations
n = Total number of items
r = Number of items selected
! = Factorial (e.g., 5! = 5x4x3x2x1 = 120)

How to Calculate Permutations (Step-by-Step)

1

Identify n and r

Determine the total number of items (n) and how many you're selecting (r). Example: Choosing 3 letters from A, B, C, D, E means n=5 and r=3.

2

Calculate n factorial (n!)

Multiply n by all positive integers less than it: 5! = 5 x 4 x 3 x 2 x 1 = 120

3

Calculate (n-r) factorial

Subtract r from n and find that factorial: (5-3)! = 2! = 2 x 1 = 2

4

Divide n! by (n-r)!

P(5,3) = 120 / 2 = 60 permutations

Permutation vs. Combination: What's the Difference?

The key difference is whether order matters:

Feature Permutation Combination
Order Matters (ABC != CBA) Doesn't matter (ABC = CBA)
Formula n! / (n-r)! n! / (r!(n-r)!)
P(5,3) vs C(5,3) 60 10
Use Case Rankings, passwords, sequences Groups, teams, committees
Example Arranging books on a shelf Choosing books to read

Pro Tip: Remember the Difference

Permutation = Position matters. Think "P for Position." If you're assigning specific roles or ranks (1st place, 2nd place, etc.), use permutations. If you're just picking items with no specific order, use combinations.

Common Applications of Permutations

  • Password generation: How many 4-digit PINs are possible? P(10,4) = 5,040
  • Race rankings: Different ways to award gold, silver, bronze
  • Scheduling: Arranging meetings in time slots
  • Seating arrangements: Arranging people in chairs
  • License plates: Combinations of letters and numbers in specific positions
  • Tournament brackets: Ordering teams in elimination rounds

Special Cases

  • P(n,n) = n! - Arranging all items (full permutation)
  • P(n,1) = n - Selecting just one item
  • P(n,0) = 1 - Selecting nothing (one way to do nothing)
  • 0! = 1 - By definition, zero factorial equals one