Permutation without Repetition Calculator

Calculate permutations nPr where order matters and elements cannot be repeated

Calculate Permutations without Repetition

Total number of distinct objects available

Number of objects to arrange in order

Number of Permutations

720
P(10,3) = 720
Permutations without repetition

Formula

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

Where n = 10 (total objects) and r = 3 (objects to arrange)

P(10,3) = 10! / (7)!

P(10,3) = 36,28,800 / 5,040 = 720

Example Calculation

Contest Ranking Problem

Problem: In a contest with 10 participants, how many ways can we arrange the top 3 winners (1st, 2nd, 3rd place)?

Given: n = 10 participants, r = 3 positions

Note: Order matters (1st ≠ 2nd ≠ 3rd), no repetition allowed

Solution

P(10,3) = 10! / (10-3)! = 10! / 7!

P(10,3) = 10 × 9 × 8

P(10,3) = 720 ways

There are 720 different ways to arrange the top 3 winners from 10 contestants.

Key Concepts

Without Repetition

Each element can only be used once

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

Order Matters

ABC ≠ ACB ≠ BAC

Different arrangements = different permutations

Sequential Selection

First: n choices, Second: n-1 choices

Pattern: n × (n-1) × ... × (n-r+1)

Quick Examples

P(5,2)20
P(7,3)210
P(10,4)5,040
P(8,8)40,320

Mathematical Properties

P(n,0) = 1 (empty arrangement)

P(n,1) = n (single element)

P(n,n) = n! (all elements)

P(n,r) = C(n,r) × r!

Understanding Permutations without Repetition

What are Permutations without Repetition?

A permutation without repetition is an arrangement of distinct objects where the order matters and each object can only be used once. When you select r objects from n distinct objects and arrange them in a specific order, you're calculating P(n,r) or nPr.

Key Characteristics

  • Order matters: ABC ≠ BAC ≠ CAB
  • No repetition: Each element used only once
  • Selection and arrangement: Choose r from n and order them
  • Sequential choices: n choices first, n-1 second, etc.

Common Applications

  • Arranging people in a line or seating order
  • Contest rankings (1st, 2nd, 3rd place)
  • Password creation (without repeated characters)
  • License plate combinations
  • Committee officer positions
  • Scheduling distinct tasks in order

Memory Tip: Think "Position matters, No repeats!" - each position gets a unique item, and switching positions creates a different arrangement.

Mathematical Formula

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

Permutations of r objects from n distinct objects

Where:

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

Alternative Formula:

  • P(n,r) = n × (n-1) × ... × (n-r+1)
  • Multiply r consecutive decreasing integers
  • More efficient for large n, small r
  • Avoids calculating large factorials

Example: Race Positions

Problem: 8 runners in a race. How many ways can 1st, 2nd, and 3rd place be awarded?

Solution: P(8,3) = 8!/(8-3)! = 8!/5! = 8×7×6 = 336 ways

Example: Password Creation

Problem: Create a 4-digit password using digits 1-9, no repetition allowed.

Solution: P(9,4) = 9!/(9-4)! = 9!/5! = 9×8×7×6 = 3,024 passwords