Permutation & Combination Calculator
Calculate nPr (permutations) and nCr (combinations) with step-by-step factorial breakdowns. Everything runs in your browser — nothing is stored or sent to any server.
How Permutation & Combination Calculator Works
Calculate permutations and combinations instantly with step-by-step factorial breakdowns. runs in your browser. Enter your values into the form above and the calculator processes them instantly in your browser — no data is sent to any server.
Understanding Permutations and Combinations
Permutations and combinations are fundamental counting techniques in mathematics used to determine the number of possible arrangements or selections from a set of items. They appear everywhere — from probability and statistics to cryptography, genetics, and game theory.
The key distinction is straightforward: permutations care about order, while combinations do not. Choosing Alice, Bob, and Carol for president, vice-president, and treasurer is a permutation problem (ABC differs from BAC). Choosing 3 people for a committee is a combination problem (the same group regardless of order).
Formulas
Permutation: nPr = n! / (n - r)!
Combination: nCr = n! / [r! × (n - r)!]
Where:
- n = total number of items
- r = number of items chosen
- n! = n factorial = n × (n-1) × (n-2) × ... × 1
Real-World Examples
- Lottery numbers: Picking 6 numbers from 49 is a combination problem — order does not matter. C(49,6) = 13,983,816 possible tickets.
- Team selection: Choosing 5 starters from 12 basketball players is C(12,5) = 792 possible lineups.
- Passwords: Arranging 4 digits from 0-9 without repetition is P(10,4) = 5,040 passwords.
- Race rankings: Gold, silver, bronze from 8 runners is P(8,3) = 336 possible podiums.
Permutation vs Combination: Quick Rule
Ask yourself: "Does the order in which I pick items change the outcome?" If yes, use permutations. If no, use combinations. For instance, a PIN code 1234 is different from 4321 (permutation), but a hand of cards {A, K, Q} is the same regardless of draw order (combination).
Factorials Explained
A factorial (written as n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1. Factorials grow extremely fast — 20! is already over 2.4 quintillion. This calculator handles values up to n = 170 before JavaScript number precision limits are reached.
Applications
Combinatorics underpins probability theory, statistical sampling, algorithm design (sorting, hashing), error-correcting codes, DNA sequencing, and tournament scheduling. Understanding permutations and combinations is essential for standardised tests like the GRE, GMAT, SAT, and competitive programming contests.