Permutation & Combination Calculator

About the Permutation & Combination Calculator

A permutation and combination calculator computes the number of ways to arrange or select items from a set — the foundational counting principles of combinatorics and probability. Permutations count ordered arrangements where sequence matters (arranging 3 books from a shelf of 10 in a specific order). Combinations count unordered selections where only membership matters (choosing 3 committee members from 10 people, regardless of who was chosen "first"). These calculations appear in probability theory, lottery odds, password security analysis, scheduling problems, genetics (gene combinations), experiment design, and the analysis of card games and board games. Our calculator handles both P(n,r) and C(n,r) for any n and r, with and without repetition, shows the step-by-step factorial computation, and also calculates binomial coefficients, multinomial coefficients, and circular permutations.

How It Works

Permutation (ordered, no repetition): P(n,r) = n! / (n-r)!. The number of ways to arrange r items from n distinct items in order. Example: arranging 4 books from a shelf of 9: P(9,4) = 9! / 5! = 9 x 8 x 7 x 6 = 3,024 arrangements. Combination (unordered, no repetition): C(n,r) = n! / (r! x (n-r)!). Choosing 4 books from 9 to take on vacation: C(9,4) = 9! / (4! x 5!) = 126 selections. Relationship: C(n,r) = P(n,r) / r! — combinations are permutations divided by the r! ways to arrange the selected items (which we don't care about in combinations). With repetition: Permutations with repetition = n^r. Combinations with repetition = C(n+r-1, r).

Tips & Best Practices

• ✓Key distinction: "arrangement" or "order matters" implies permutation; "selection," "group," or "committee" implies combination.
• ✓Symmetry property: C(n,r) = C(n, n-r). Choosing 3 from 10 gives the same count as choosing 7 from 10 — you are implicitly choosing which 7 NOT to take.
• ✓Lottery odds (Powerball): C(69,5) = 11,238,513 ways to choose the 5 white balls, times 26 Powerball options = 292,201,338:1 odds. The combination calculator reveals the true scale of lottery improbability.
• ✓Password security: an 8-character password using 94 printable ASCII characters has 94^8 ≈ 6.1 x 10^15 combinations with repetition — practically impossible to brute-force even with modern computers.
• ✓Circular permutations: arranging n objects in a circle = (n-1)! because rotations of the same arrangement are considered identical. Arranging 8 people at a round table: 7! = 5,040 distinct arrangements.
• ✓Binomial theorem: (a+b)^n = sum of C(n,k) x a^(n-k) x b^k for k from 0 to n. The coefficients C(n,k) are the binomial coefficients forming Pascal's triangle.
• ✓Identical items: if some items are identical, divide by the factorial of the count of each identical item. Arrangements of letters in "MISSISSIPPI": 11! / (4! x 4! x 2!) = 34,650.
• ✓Derangements: permutations where no element appears in its original position. Number of derangements of n items = n! x sum((-1)^k / k!) for k=0 to n, approximately n!/e.

Who Uses This Calculator

Statistics and probability students solving counting problems. Lottery players calculating true odds. Security analysts determining password strength. Biologists counting gene combinations and DNA sequence possibilities. Game designers calculating deck combinations and probability distributions. Operations researchers solving assignment and scheduling problems. Teachers creating probability word problems with known correct answers.