Permutation Calculator

Calculate Permutation P(n, r)

A permutation counts the number of ways to arrange r items from a set of n items where order matters.

Quick Examples

🏆 Race Podium

Gold, Silver, Bronze from 10 athletes

P(10, 3) = 720

🔤 3-Letter Code

Using 26 letters (no repeats)

P(26, 3) = 15,600

🔢 4-Digit PIN

From digits 1-9 (no repeats)

P(9, 4) = 3,024

👥 Queue of 8

Arrange 8 people in a line

P(8, 8) = 8! = 40,320

🃏 5-Card Sequence

Ordered 5 cards from deck

P(52, 5) = 311M+

📊 Officer Election

Pres, VP, Sec, Treas from 20

P(20, 4) = 116,280

Permutation Formula

The Formula

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

Or equivalently, multiply the first r terms of n!:

P(n, r) = n × (n-1) × (n-2) × ... × (n-r+1)

Why This Works

First position: n choices
Second position: n-1 choices (one item used)
Third position: n-2 choices
...and so on until r positions are filled

Special Cases

Case	Value	Explanation
P(n, 0)	1	One way to arrange zero items (do nothing)
P(n, 1)	n	n ways to pick one item
P(n, n)	n!	Arrange all n items = n factorial
P(n, n-1)	n!	Same as P(n,n) — last item is determined

Permutation vs Combination

Key difference: Does order matter?

	Permutation P(n,r)	Combination C(n,r)
Order	Matters ✓	Doesn't matter
Formula	n! / (n-r)!	n! / (r! × (n-r)!)
Example	ABC ≠ BCA ≠ CAB	{A,B,C} = {B,C,A} = {C,A,B}
P(10,3) vs C(10,3)	720	120
Relationship	P(n,r) = C(n,r) × r!

When to Use Each

Scenario	Use	Why
Lottery numbers	Combination	Numbers aren't ordered
Password/PIN	Permutation	1234 ≠ 4321
Committee selection	Combination	Members are equal
Race finish order	Permutation	1st, 2nd, 3rd are different
Poker hand	Combination	Cards can be reordered
Book arrangement	Permutation	Left-to-right order matters

Factorial Reference

n	n!	Calculation
0	1	By definition
1	1	1
2	2	2 × 1
3	6	3 × 2 × 1
4	24	4 × 3 × 2 × 1
5	120	5 × 4 × 3 × 2 × 1
6	720	6!
7	5,040	7!
8	40,320	8!
9	362,880	9!
10	3,628,800	10!
12	479,001,600	~479 million
15	1.307 × 10¹²	~1.3 trillion
20	2.432 × 10¹⁸	~2.4 quintillion

Privacy & Limitations

All calculations run entirely in your browser -- nothing is sent to any server.
Results are computed using standard formulas and should be verified for critical applications.

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Permutation Calculator FAQ

What is Permutation Calculator?

Permutation Calculator is a free math tool that helps you Calculate nPr permutations with step-by-step formula breakdown.

How do I use Permutation Calculator?

Enter your input values, review the calculated output, and adjust inputs until you reach the result you need. The result updates in your browser.

Is Permutation Calculator private?

Yes. Calculations run locally in your browser. Inputs are not uploaded to a server by default, and refreshing the page clears session data.

Does Permutation Calculator require an account or installation?

No. You can use this tool directly in your browser without sign-up or software installation.

How accurate are results from Permutation Calculator?

This tool applies standard formulas or deterministic processing logic for estimates. For medical, legal, tax, or investment decisions, verify with a qualified professional.

Can I save or share outputs from Permutation Calculator?

You can bookmark this page and copy outputs manually. Results are not persisted in your account and are typically not embedded in the URL.