Compute permutations (nPr), combinations (nCr) and factorial (n!) instantly. The calculator uses BigInt so even huge values like 100C50 give an exact integer answer instead of overflowing. Useful for board exam probability questions, lottery odds, password strength estimation and combinatorial puzzles.
15C11 = 15! / (11! × 4!)
1,365
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Permutations and combinations are foundational counting tools in mathematics, probability, statistics and computer science. They answer 'how many ways' questions: how many ways to arrange 11 players in batting order from a 15-member squad (a permutation), how many ways to pick 6 lottery numbers from 49 (a combination), how many distinct arrangements of the letters in MISSISSIPPI exist (a permutation with repetitions). Class 11-12 mathematics has an entire chapter on this topic; JEE and other competitive exams test it heavily.
This calculator uses BigInt arithmetic internally so it returns exact integer answers even for inputs that would normally overflow JavaScript's standard number type. 21! already exceeds 2^53, the limit of regular JavaScript numbers; 100C50 is a 30-digit integer. BigInt computes both exactly, which matters for cryptography, combinatorial proofs, and any context where you need the exact integer count rather than an approximation.
Pick the operation: nPr (order matters), nCr (order does not matter), or n! (factorial).
Enter n - the total number of items.
If picking nPr or nCr, enter r - the number of items chosen.
The answer appears instantly with the formula and step-by-step expansion.
Permutations count arrangements where order matters - for example, ranking the top 3 students in a class of 30. Combinations count selections where order doesn't matter - for example, choosing 11 cricketers from a squad of 15. Factorial counts the total arrangements of n distinct objects: n! = n × (n−1) × ... × 1. All three are linked by simple formulas, and our implementation uses BigInt internally so values like 200! return an exact integer rather than ∞.
nPr = n! / (n − r)! nCr = n! / (r! × (n − r)!)
Factorial: n! = n × (n−1) × (n−2) × ... × 2 × 1 counts the total ways to arrange n distinct objects in order. By convention, 0! = 1 (which makes downstream formulas like nC0 = 1 work cleanly). Factorials grow extremely fast - 10! = 3.6 million, 20! ≈ 2.4 × 10^18, 100! has 158 digits. BigInt handles these without overflow.
Permutations: nPr = n!/(n−r)! counts the number of ordered arrangements of r items chosen from n. The formula derives from picking the first slot (n choices), then the second (n−1 choices), and so on - multiplying gives n × (n−1) × ... × (n−r+1), which equals n!/(n−r)!. Order matters: arranging A-B-C is different from B-A-C.
Combinations: nCr = n!/(r!(n−r)!) counts the number of unordered selections of r items from n. Each combination of r items can be arranged in r! different orders, so nCr = nPr / r!. Order doesn't matter: choosing {A, B, C} is the same as choosing {B, C, A}. The notation nCr also written as C(n,r) or (n choose r) - Pascal's triangle is literally a triangular display of all the nCr values.
15C11 = 15!/(11!×4!) = 1,365 distinct teams. If team batting order matters (which usually it does): 15P11 = 15!/4! = 217,945,728,000 - over 217 billion ordered arrangements. The permutation count explains why team management is a real strategic challenge.
Total combinations: 49C6 = 49!/(6!×43!) = 13,983,816. Probability of winning with a single ticket: 1 in ~14 million. To put it in perspective - you're roughly 5 times more likely to be struck by lightning than win the jackpot with one ticket.
If characters can repeat: 62^8 = 2.18 × 10^14 = 218 trillion possibilities. If no repetition: 62P8 = 1.36 × 10^14. Either way, brute-force on modern hardware would take hours-days for the first, weeks-months for the second.
The number of ways to arrange r items chosen from n distinct items, where order matters. Example: how many ways can 3 people stand in a line picked from 10 candidates? 10P3 = 10! / 7! = 720.
The number of ways to pick r items from n where order does not matter. Example: choosing 11 players from a squad of 15 → 15C11 = 1,365 different teams.
nPr always counts more (or equal) than nCr because each combination of r items can be ordered in r! different ways. So nPr = nCr × r!.
0! = 1 by convention. This makes formulas like nC0 = 1 (there's exactly one way to pick nothing) work cleanly.
JavaScript's regular numbers lose precision past 2⁵³. 21! already overflows past that. With BigInt, we compute exactly even for 1000! - the digits are correct.
No. Permutations and combinations are defined only for non-negative integers, and r ≤ n. The calculator flags inputs outside this range as invalid.
Pick 6 numbers from 49: 1 / 49C6 = 1 / 13,983,816 - about 1 in 14 million. The combinations matter, not the order. Use 49C6 in this calculator.
Each entry in Pascal's triangle is exactly nCr. Row n has entries nC0, nC1, ..., nCn. They sum to 2ⁿ - the total number of subsets of an n-element set.
From simple percentages to scientific notation, statistics, geometry and big-number arithmetic - every common math problem you'd reach for a calculator for, runs in your browser with no installs. The fraction calculator preserves exact rationals, the big-number calculator uses BigInt for hundreds of digits without precision loss, and the scientific calculator's expression evaluator is a hand-written shunting-yard parser (no eval) that supports trig, log, factorial and parentheses.
If you're a student, double-check your work without paying for a graphing calculator. If you're a professional, use these tools as a sanity check on spreadsheet formulas. Every result is shown alongside its formula and inputs, so the math is always auditable.
The Permutation & Combination Calculator is part of the Math suite on Online Calculator Tools, alongside 6 related calculators. All tools share the same UI patterns, the same privacy-first rendering (nothing is sent to a server), and the same commitment to showing the formula and assumptions behind every result.
