Permutation Calculator

Calculate the number of possible arrangements P(n,r) with step-by-step solutions

Input Values

Result

What are Permutations?

A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time, denoted as P(n,r), represents the number of different ways to arrange r objects from a set of n distinct objects where order matters.

Permutation Formula

The formula for calculating permutations is:

P(n,r) = n! / (n-r)!

Where:

  • n = Total number of objects
  • r = Number of objects to arrange
  • n! = n factorial (n × (n-1) × (n-2) × ... × 1)
  • (n-r)! = (n-r) factorial

When to Use Permutations

  • When order matters (e.g., arranging people in a line)
  • When selecting and arranging objects
  • For password combinations
  • For race finishing orders
  • For seating arrangements

How to Use the Calculator

  1. Enter the total number of objects (n)
  2. Enter the number of objects to arrange (r)
  3. Click "Calculate" to get the number of permutations
  4. View the step-by-step solution

Example Calculations

P(5,3):

n = 5, r = 3

P(5,3) = 5! / (5-3)!

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

P(4,4):

n = 4, r = 4

P(4,4) = 4! / (4-4)!

P(4,4) = 4! / 0! = 24 / 1 = 24

Important Notes

  • r must be less than or equal to n
  • 0! = 1 (by definition)
  • P(n,n) = n! (all objects arranged)
  • P(n,1) = n (one object selected)
  • P(n,0) = 1 (no objects arranged)

Difference Between Permutations and Combinations

Permutations (P(n,r)):

Order matters

ABC ≠ BAC

Formula: n! / (n-r)!

Combinations (C(n,r)):

Order doesn't matter

ABC = BAC

Formula: n! / (r! × (n-r)!)