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.

Permutation (nPr)

Result

n (Total Items)
0
r (Items to Choose)
0
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
5P3 = 60

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.

A permutation is an arrangement of items where order matters. For example, the arrangements ABC, ACB, BAC, BCA, CAB, CBA are all different permutations of the letters A, B, and C. Permutations count the number of ways to arrange r items from n total items.

The permutation formula is nPr = n! / (n-r)!, where n is the total number of items and r is the number of items to arrange. For example, 5P3 = 5! / (5-3)! = 120 / 2 = 60.

Permutations consider order (ABC is different from CBA), while combinations ignore order (ABC and CBA are the same combination). Permutations give more arrangements than combinations for the same n and r values.

Use permutations when order matters: arranging people in a line, creating passwords, ranking items, assigning positions, or any situation where the sequence is important.

No, r cannot be greater than n. You cannot arrange more items than you have. For example, you cannot arrange 5 items from a set of 3. The calculator will show an error if r > n.

When r = n, you're arranging all items. In this case, nPr = n! (n factorial). For example, 5P5 = 5! = 120, which is the number of ways to arrange all 5 items.