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Permutation Calculator

Calculate the number of ways to arrange r items from a set of n items where order matters. Uses the formula P(n,r) = n!/(n-r)!

Reviewed by Christopher FloiedPublished Updated

This free online permutation calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

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Results

Number of Permutations P(n,r)

720

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Permutation Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Permutation Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Permutation Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About Permutation Calculator

The Permutation Calculator solves one of the fundamental questions in combinatorics: 'In how many different ways can I arrange r items from a set of n?' Unlike combinations, order matters in permutations — ABC is different from BCA. This is exactly what you need when the sequence of selections affects the outcome: ranking contestants, selecting officers in order (president, VP, secretary), arranging books on a shelf, creating passwords, or choosing a podium for a race. Permutations appear in probability calculations, cryptography, game theory, algorithm analysis, and statistics. Understanding P(n,r) is the gateway to combinatorial reasoning and helps you count possibilities in complex scenarios.

The Math Behind It

A permutation is an arrangement of objects where the order of selection matters. The number of ways to arrange r items from n distinct items is denoted P(n,r) or nPr. **The Formula**: P(n,r) = n! / (n-r)! Where: - n! = n factorial (n × (n-1) × (n-2) × ... × 1) - (n-r)! = the factorial of the items NOT chosen **Intuition**: For r items chosen from n: - First choice: n options - Second choice: n-1 options (one already taken) - Third choice: n-2 options - ... - r-th choice: n-r+1 options Total: n × (n-1) × (n-2) × ... × (n-r+1) = n!/(n-r)! **Key Distinction from Combinations**: - **Permutation**: Order matters — ABC ≠ BCA - **Combination**: Order doesn't matter — ABC = BCA For the same n and r, permutations = r! × combinations. **Common Applications**: 1. **Racing**: In how many ways can 3 of 8 runners finish on the podium? P(8,3) = 8!/5! = 336 2. **Passwords**: How many 4-character passwords with distinct letters from 26? P(26,4) = 26!/22! = 358,800 3. **Seating arrangements**: How many ways to seat 5 people at 5 chairs? P(5,5) = 5! = 120 4. **Tournament rankings**: Top 3 finishers among 10 teams? P(10,3) = 10!/7! = 720 **Special Cases**: - **P(n,n) = n!**: Permutations of all items (full arrangement) - **P(n,0) = 1**: Arrange nothing — only one way (empty) - **P(n,1) = n**: Choose one of n — n ways - **P(n,r) > C(n,r)**: Always more permutations than combinations for r > 1 **Permutations with Repetition**: If items can repeat (like in passwords allowing reuse), the formula is simply n^r. For example, 4-character passwords with letters that can repeat: 26^4 = 456,976. **Permutations of Identical Items**: When some items are identical, divide by the factorial of each group of identicals. For 'MISSISSIPPI' (11 letters: 1M, 4I, 4S, 2P): Arrangements = 11! / (1! × 4! × 4! × 2!) = 34,650 **Circular Permutations**: Arranging n people around a round table: (n-1)! (one person's position is fixed; others rotate around them). **Factorial Growth**: | n | n! | |---|-----| | 5 | 120 | | 10 | 3,628,800 | | 15 | 1.31 × 10¹² | | 20 | 2.43 × 10¹⁸ | | 25 | 1.55 × 10²⁵ | Factorials grow astronomically — even modest n gives unimaginable numbers.

Formula Reference

Permutation Formula

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

Variables: n = total items, r = items chosen, order matters

Worked Examples

Example 1: Race Podium

In a race of 10 runners, how many ways can runners finish in the top 3 positions (gold, silver, bronze)?

Step 1:n = 10, r = 3
Step 2:P(10,3) = 10! / 7!
Step 3:P(10,3) = (10 × 9 × 8 × 7!) / 7!
Step 4:P(10,3) = 10 × 9 × 8
Step 5:P(10,3) = 720

720 different ordered podium outcomes. If it were a combination (just 'top 3' without order), it would be 720/3! = 120.

Example 2: Unique Passwords

A 5-character password uses distinct letters from the 26-letter alphabet. How many possible passwords?

Step 1:n = 26, r = 5
Step 2:P(26,5) = 26!/21!
Step 3:P(26,5) = 26 × 25 × 24 × 23 × 22
Step 4:P(26,5) = 7,893,600

Nearly 8 million possible passwords. If repetition were allowed (26^5), you'd have 11,881,376 — about 50% more.

Common Mistakes & Tips

  • !Confusing permutations with combinations. If order matters, use permutations; if not, use combinations.
  • !Forgetting that P(n,n) = n! (all items arranged). This is the factorial special case.
  • !Computing (n-r)! wrong. Remember (n-r)! is the factorial of the count of items you DIDN'T choose.
  • !Applying permutation formula when items can repeat. With repetition, use n^r instead.

Related Concepts

Combination Calculator

When order does NOT matter — C(n,r) = P(n,r)/r!.

Factorial Calculator

The core of permutation math.

Probability

Permutations often show up in probability calculations.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Combination Calculator

Frequently Asked Questions

When should I use permutations vs combinations?

Ask: 'Does order matter?' If yes (first place, second place, third place; positions; rankings; sequences), use permutations. If no (just 'which 3 were chosen', 'committee of 5', 'hand of cards'), use combinations. Example: picking a committee of 3 from 10 people = combination (120 ways). Picking president, VP, and treasurer from 10 = permutation (720 ways).

What's the relationship between permutations and combinations?

P(n,r) = C(n,r) × r!. In other words, permutations = combinations × arrangements of chosen items. If you pick 3 people from 10 as a 'team' (combination: 120 ways), and then arrange those 3 in a specific order (3! = 6 ways), you get 120 × 6 = 720 permutations.

Can r be larger than n?

Not in standard permutations (without repetition). You can't arrange more items than you have. P(5,7) is undefined — you can't pick 7 items from a set of 5 unique items. If repetition is allowed, then you can have r > n, and the count is simply n^r (each position has n choices).

How are permutations used in real life?

Everywhere order matters: race results (first, second, third), lottery Pick-3 games, lineup orders in baseball, password generation, seating arrangements, tournament brackets, DNA sequence analysis, scheduling problems, routing optimization (like Travelling Salesman), playlist ordering, and cipher cryptography. Even shuffling a deck of cards uses 52! permutations — so many that every shuffle has likely never been seen before.