Permutationsrechner

What is a permutation?

A permutation is an arrangement of items where order matters. If you're picking who finishes first, second, and third in a race, the arrangement (Alice, Bob, Carol) is different from (Carol, Bob, Alice) — same three people, but the finishing order is different. Each ordering is a separate permutation.

The formula for P(n, k) — the number of permutations of k items chosen from n distinct items — is:

Read this as "n permute k". The product version is k terms long: you start at n and multiply by one less each time, k times. For P(5, 3): 5 × 4 × 3 = 60. For P(10, 4): 10 × 9 × 8 × 7 = 5040. The first position has n choices, the second has n − 1 (since one item is already placed), and so on.

How to use the permutation calculator

1. Enter n — the total number of distinct items you have.
2. Enter k — the number of positions you're filling. k must be at most n; you can't put more items in positions than you have items.
3. The result P(n, k) appears instantly with the formula displayed.
4. For very large results (more than 30 digits), the calculator switches to scientific notation (e.g., 1.234 × 10²⁰).
5. Tap a worked example to load classic permutation problems — podium finishes, lottery-style ordered draws, code arrangements.

Permutations vs combinations — the critical distinction

This is the single most-asked question in combinatorics. The rule:

• Permutations (P): order matters. Gold-Silver-Bronze ≠ Bronze-Silver-Gold. Three positions, three people.
• Combinations (C): order doesn't matter. {Alice, Bob, Carol} is one combination regardless of how you list them.

The relationship between them:

Why? Because every combination of k items can be arranged in k! different orders. C(5, 3) = 10 (the number of ways to choose 3 from 5 ignoring order); P(5, 3) = 60 = 10 × 3! = 10 × 6 (each of those 10 combinations can be arranged 6 different ways).

Quick test: "Does the order matter?" If yes → permutations. If no → combinations.

Worked examples

Example 1 — Podium finishes

10 athletes are competing for gold, silver, and bronze. How many possible podium results?

P(10, 3) = 10 × 9 × 8 = 720. The order matters because gold ≠ silver ≠ bronze. (If you only cared about which 3 athletes made the podium regardless of position, you'd want C(10, 3) = 120.)

Example 2 — Bookshelf arrangement

You have 5 books and you want to know how many different ways you could arrange them on a shelf.

P(5, 5) = 5 × 4 × 3 × 2 × 1 = 120. When k = n, P(n, n) = n! — the basic factorial. Every full arrangement of all 5 books is a permutation.

Example 3 — Three-letter codes (no repeats)

How many 3-letter codes can you make from the 26 English letters if no letter can repeat?

P(26, 3) = 26 × 25 × 24 = 15,600. The first letter has 26 choices; once placed, the second has 25 (any letter except the first); the third has 24.

Note: if letters CAN repeat, the answer is 26³ = 17,576 (independent choices). The 1,976 difference is the codes with at least one repeated letter.

Example 4 — Five-card sequences

From a 52-card deck, how many ordered sequences of 5 cards can you draw (where the order you draw them in matters)?

P(52, 5) = 52 × 51 × 50 × 49 × 48 = 311,875,200. About 312 million ordered sequences. If you don't care about order (just the 5-card hand), it's C(52, 5) = 2,598,960 — the famous "five-card hand" number used in poker probability.

Example 5 — Top-4 finishers from 100

P(100, 4) = 100 × 99 × 98 × 97 = 94,109,400. About 94 million possible top-4 outcomes from a field of 100. Useful for ranking-prediction probability calculations.

Special cases

P(n, n) = n!

When you arrange all items, you get a plain factorial. P(5, 5) = 5! = 120. P(10, 10) = 10! = 3,628,800.

P(n, 0) = 1

There's exactly one way to arrange zero items: the empty arrangement. This convention matches 0! = 1.

P(n, 1) = n

If you're filling just one position from n options, you have n choices. Trivially the same as just "pick one of n."

P(n, 2) = n(n−1)

Two positions, n choices for the first, (n−1) for the second. Useful shortcut: P(10, 2) = 10 × 9 = 90 ordered pairs from 10 items.

Permutations with repetition

The basic P(n, k) formula assumes WITHOUT repetition — each item is used at most once. If items can repeat (like digits in a PIN, characters in a password, results of a die roll), the math is simpler:

For a 4-digit PIN (10 digits, each position independent): 10⁴ = 10,000. For a 6-character password using 26 letters, 10 digits, and 30 special characters (all repeatable): 66⁶ = 82.7 billion.

This calculator uses the no-repeat formula. If you need with-repetition, just compute nᵏ in any calculator.

Permutations of multisets

What if you have repeated items in your set? Like the letters in MISSISSIPPI?

For a multiset of size n with k₁ copies of type 1, k₂ copies of type 2, etc., the number of distinct arrangements is:

For MISSISSIPPI (11 letters: 1 M, 4 I's, 4 S's, 2 P's): 11! / (1! × 4! × 4! × 2!) = 39,916,800 / 1152 = 34,650 distinct arrangements. The calculator doesn't handle multisets directly — for those, work it out by hand or use a dedicated multiset permutation tool.

Where permutations appear in the wild

Sports and competitions

Race finish orders, tournament seeding, leaderboard positions. The number of possible outcomes for a tournament of N teams is enormous: a 64-team single-elimination bracket has 2⁶³ ≈ 9.2 quintillion possible bracket fills.

Cryptography and security

Password complexity is fundamentally a permutation count. A 12-character password with 95 possible characters per position has 95¹² ≈ 5.4 × 10²³ possible passwords. PINs, passcodes, and master keys all reduce to "how many ordered arrangements?"

Scheduling

"How many ways could we schedule 8 meetings into 8 time slots?" P(8, 8) = 40,320. For a partial fill ("8 meetings into 5 of 8 slots"), it's P(8, 5) = 6720.

Genetics and biology

DNA sequences, protein folding orders, gene arrangements. The number of possible orderings of 20 amino acids in a 100-residue protein is 20¹⁰⁰ ≈ 10¹³⁰ — vastly more proteins than have ever existed or could exist.

Lottery and ordered drawings

Most lotteries are combinations (order doesn't matter when you check a winning ticket), but ordered draws — like the Super Bowl coin toss followed by a kickoff — are permutations.

Algorithms and computer science

Brute-force permutation enumeration is O(n!) — quickly intractable past n = 10. The traveling salesman problem in its naive form requires checking all (n−1)!/2 city orderings; for n = 20 that's 60 quadrillion routes.

Common mistakes

• Confusing P with C. The single most common error. Always ask: "Does order matter?" If yes, permutations; if no, combinations.
• Using P(n, k) when items can repeat. P(n, k) assumes no repetition. If items can be reused (PIN digits, dice rolls), use nᵏ.
• Forgetting that k ≤ n. You can't pick more items than you have. P(5, 7) is undefined.
• Treating P(n, 1) as 1. P(n, 1) = n, not 1. Filling one position from n distinct items gives n possible arrangements.
• Off-by-one in the product. P(n, k) is k terms long: n × (n−1) × ... × (n−k+1). Stop at (n−k+1), not (n−k). For P(5, 3): 5 × 4 × 3, not 5 × 4 × 3 × 2.

What the calculator gives you, summarized

• Exact P(n, k) — computed via BigInt for precision up to n = 1000.
• Scientific approximation — for results with more than ~30 digits, displayed alongside or instead of the exact value.
• Formula display — the n! / (n − k)! form shown with your numbers plugged in, for easy verification.
• Worked examples — podium finishes, code arrangements, card hands all loadable with one tap.
• Permutations-vs-combinations callout — the most-confused distinction in combinatorics, called out inline.

Two inputs (n and k), one BigInt-precise output. The math is just multiplication, but the answers grow alarmingly fast.