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:
P(n, k) = n! / (n − k)! = n × (n − 1) × (n − 2) × ... × (n − k + 1)
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
- Enter n — the total number of distinct items you have.
- 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.
- The result P(n, k) appears instantly with the formula displayed.
- For very large results (more than 30 digits), the calculator switches to scientific notation (e.g., 1.234 × 10²⁰).
- 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:
P(n, k) = C(n, k) × k!
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:
P with repetition = nᵏ
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:
n! / (k₁! × k₂! × ... × kₘ!)
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.