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Compute permutations (nPr, ordered arrangements) and combinations (nCr, unordered selections) using factorials — ideal for probability, lottery odds, poker hands, and password strength analysis.
Math
Generated on May 23, 2026
Compute permutations (nPr, ordered arrangements) and combinations (nCr, unordered selections) using factorials — ideal for probability, lottery odds, poker hands, and password strength analysis.
A permutation and combination calculator computes the number of ways to arrange or select r items from a set of n. Permutations (nPr) count ordered arrangements — first, second, third place on a podium — while combinations (nCr) count unordered selections — which 3 people are on a committee. These are the foundational building blocks of probability, combinatorics, and discrete mathematics, and they appear everywhere: lottery odds, card-game probabilities, password strength, statistical sampling, and genetics.
Formula
nPr = n! / (n − r)!. nCr = n! / [r! × (n − r)!]. nCr = nPr / r!.Lottery question: 'pick 6 numbers from 1 to 49 — what are my odds?' Or poker question: 'how many five-card hands can be dealt from a standard 52-card deck?' Or the question that keeps cryptographers up at night: 'how many ways can you arrange a 16-character password?' All three are combinatorics — counting possibilities — and all three reduce to permutations (when order matters) or combinations (when it doesn't). The clearest test: would 'Alice, Bob, Carol' be a different answer from 'Bob, Carol, Alice' in your problem? If yes, permutation. If no, combination. Pakistan lottery odds for 6 of 49 are 1 in 13,983,816 — and yes, that's why almost nobody wins. Poker hands? 52C5 = 2,598,960. Royal flush odds? 4 out of those nearly 2.6 million, or roughly 1 in 650,000 hands. This calculator handles nPr (ordered) and nCr (unordered) with full factorial precision, useful for school combinatorics homework, probability courses, password-strength analysis, and any back-of-envelope question about 'how many ways can this happen?' Worth knowing: factorials grow ferociously — 20! is over 2.4 quintillion, and most pocket calculators give up around 60!. JavaScript number precision starts losing accuracy around 18 digits, which limits exact calculations to roughly n = 20 or so.
A permutation and combination calculator computes the number of ways to arrange or select r items from a set of n. Permutations (nPr) count ordered arrangements — first, second, third place on a podium — while combinations (nCr) count unordered selections — which 3 people are on a committee. These are the foundational building blocks of probability, combinatorics, and discrete mathematics, and they appear everywhere: lottery odds, card-game probabilities, password strength, statistical sampling, and genetics.
Permutations count ordered arrangements: the first spot has n choices, the second has n − 1, and so on for r spots — giving n × (n − 1) × (n − 2) × … × (n − r + 1) = n! / (n − r)!. Combinations divide by r! because we are no longer counting orderings within the chosen group. That is why nCr is always smaller than nPr (except when r = 0 or r = 1, where they are equal).
Permutations (nPr, order matters) and combinations (nCr, order doesn't) for small n and r.
| n | r=2 (nPr / nCr) | r=3 (nPr / nCr) | r=4 (nPr / nCr) | r=5 (nPr / nCr) |
|---|---|---|---|---|
| 5 | 20 / 10 | 60 / 10 | 120 / 5 | 120 / 1 |
| 6 | 30 / 15 | 120 / 20 | 360 / 15 | 720 / 6 |
| 7 | 42 / 21 | 210 / 35 | 840 / 35 | 2,520 / 21 |
| 8 | 56 / 28 | 336 / 56 | 1,680 / 70 | 6,720 / 56 |
| 9 | 72 / 36 | 504 / 84 | 3,024 / 126 | 15,120 / 126 |
| 10 | 90 / 45 | 720 / 120 | 5,040 / 210 | 30,240 / 252 |
| 20 | 380 / 190 | 6,840 / 1,140 | 116,280 / 4,845 | 1.86M / 15,504 |
| 52 | 2,652 / 1,326 | 132,600 / 22,100 | — | 311M / 2.6M (poker hands) |
52C5 = 2,598,960 — the number of unique 5-card poker hands from a standard deck.
4-digit PIN: only 10,000 combinations (10^4) — a computer cracks it in milliseconds.
8-character password (lowercase only): 26^8 ≈ 208 billion combinations.
Rubik's cube: 4.3 × 10^19 possible states — yet solvable in at most 20 moves (God's number).
Ten sprinters racing for gold, silver, and bronze: 10P3 = 720 ordered outcomes (different medals mean different outcomes).
Picking 3 people out of 10 for a committee: 10C3 = 120 combinations (committee members are unordered).
Lottery drawing 6 numbers from 49: 49C6 = 13,983,816 possible tickets — one reason winning is so unlikely.
A 4-digit PIN with no repeated digits: 10P4 = 5,040 possibilities — stronger than it sounds, but still brute-forceable.
Seating 8 dinner guests in a row: 8P8 = 8! = 40,320 different arrangements.
Poker hand: 5 cards drawn from 52: 52C5 = 2,598,960 possible hands — the denominator for all basic poker probability calculations.
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