What is a permutation?

A permutation is the number of ways to arrange r elements from n distinct elements in order, denoted P(n,r) or A(n,r), calculated as P(n,r)=n!/(n-r)!. Permutation considers the order of elements.

What is the difference between permutation and combination?

Permutation considers order, combination does not. For example, choosing 2 from 3 people: 6 permutations, 3 combinations. P(n,r)=n!/(n-r)!, C(n,r)=n!/(r!(n-r)!)

What are the applications of permutation?

Permutation is commonly used in password combination calculations, tournament scheduling, queuing problems, seating arrangements, and other scenarios where order matters.

🔢 Permutation Calculator

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

Total n

Choose r

P(10, 3)

720

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

Formula Explanation

P(n,r) represents the number of ways to arrange r of n distinct elements in order.

When r = n, P(n,n) = n!, which is a full permutation.

When r = 0, P(n,0) = 1, meaning there is exactly one empty permutation.

Permutation Calculator Guide

This calculator computes the permutation P(n,r) = n!/(n-r)!. Enter the total number n and selection number r to calculate the permutation count, supporting large numbers.

Permutations consider the order of elements — the number of ways to arrange r items from n distinct items. Permutations have important applications in cryptography, probability, and scheduling.

Permutation Formula

Permutation formula:

P(n,r) = n! ÷ (n-r)! = n × (n-1) × ... × (n-r+1)

Special cases: - P(n,0) = 1 - P(n,1) = n - P(n,n) = n! (full permutation)

Examples: - P(5,3) = 5×4×3 = 60 - P(10,4) = 10×9×8×7 = 5,040 - P(8,8) = 8! = 40,320

Applications: 1. Seating arrangements: 8 people in 8 seats = 40,320 ways 2. Password combinations: 4-digit PIN from 10 digits = P(10,4) = 5,040 3. Competition order: Performance sequence

This calculator is for reference only. Please verify results independently.

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Permutation Calculator Guide

The Permutation Calculator computes the number of ways to arrange r elements from n distinct elements in order, denoted P(n,r) or A(n,r). Unlike combinations, permutations consider the order of elements — (A,B) and (B,A) are treated as two different permutations.

Permutation problems are very common in daily life: queuing order, password combinations, tournament scheduling, seat assignments, etc. This tool supports the range 0 to 170, with results automatically formatted for readability (large numbers use scientific notation).

Permutation Formula

【Basic Formula】P(n,r) = n! / (n-r)!
= n × (n-1) × (n-2) × ... × (n-r+1)

【Special Cases】
P(n,n) = n! (Full permutation)
P(n,0) = 1   (Empty permutation)
P(n,1) = n

【Relationship with Combination】P(n,r) = C(n,r) × r!
Permutation = Combination × Full permutation of r elements

Practical Examples

📋

【Example 1】Choose 3 from 5 people to line up P(5,3) = 5! / (5-3)! = 5!/2! = 5×4×3 = 60 Explanation: 60 different ways to arrange them 【Example 2】4-digit PIN (0-9) P(10,4) = 10!/6! = 10×9×8×7 = 5040 Explanation: If no digit repeats, there are 5040 possibilities 【Example 3】8 people around a circular table Circular permutation = P(7,7)/8 = 7!/8 = 630 (Circular permutation fixes one position to eliminate rotational equivalence) 【Example 4】Top 3 possibilities in a competition 10 teams: P(10,3) = 10×9×8 = 720 possible rankings

Important Notes

💡

- r cannot be greater than n; otherwise, it's impossible to arrange r from n elements - When r=n, it's a full permutation, equal to n! - Permutation grows extremely fast: P(20,10) ≈ 6.7×10¹¹ - Permutations with repetition: n^r (e.g., 4-digit PIN has 10^4=10000 possibilities) - Circular and linear permutations differ; circular permutation divides by r

Application Scenarios

- Password security: Evaluate password strength, calculate brute force combination space
- Tournament scheduling: Round-robin and elimination match arrangements
- Lottery draws: Sequential drawing arrangement counting
- Shift scheduling: Different employee rotation plan counts
- Phone numbers: Area code + number valid combination analysis

Types of Permutation Problems

Permutation problems can be classified into: (1) Linear permutation — arranged in a line, the most common form; (2) Circular permutation — arranged in a circle, requires fixing a reference point; (3) Permutation with repeated elements — divide by factorials of repetition counts; (4) Derangement — permutations where no element appears in its original position.

For example, the full permutation of "MISSISSIPPI" (11 letters) is 11!/(4!×4!×2!) = 34,650, because M appears 1 time, I appears 4 times, S appears 4 times, and P appears 2 times. These are advanced applications of permutation calculation.

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