Written n!, it's the product of every positive integer from 1 to n. So 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! is defined as 1, which keeps formulas like nCr from breaking when r is 0 or n. Factorials grow absurdly fast — 10! is already 3.6 million, and 20! is over 2.4 quintillion. They explode the moment n gets serious.

When should I use permutation vs combination?

Ask one question: does swapping the order of the chosen items give you a different answer? If yes — race results, passwords, seating arrangements — it's a permutation. If no — committees, lottery picks, poker hands — it's a combination. The mantra is 'is ABC different from CBA in this problem?' That single question resolves 90 percent of textbook confusion.

Is nPr always larger than nCr?

Yes, whenever r is greater than 1, because nPr = nCr × r!. Every combination of r items can be rearranged in r! different orders, so permutations count each combination r! times. For r = 0 (empty selection) and r = 1 (single item), they're equal — no ordering possible when you've picked nothing or just one thing.

What about arrangements with repetition allowed?

Different formulas. Permutations with repetition (pick from n with replacement, fill r positions) = n^r — that's how PIN strength is calculated. Combinations with repetition (pick r from n with replacement, order doesn't matter) = (n + r − 1) C r. Classic example: picking 3 scoops of ice cream from 10 flavors with repeats allowed gives you (10 + 3 − 1) C 3 = 220 possible sundaes.

What are real-world applications?

Probability calculations (card games, lottery, dice), password and PIN strength analysis, cryptography, experimental design (how many variants do we need to test?), bioinformatics (protein sequence counts), quality sampling, tournament scheduling, and combinatorial optimization. Almost any problem that involves counting possibilities is secretly a permutation or combination question in disguise.

Why are 4-digit PINs weak?

Just 10,000 possibilities total (10^4, permutations with repetition). A phone trying one PIN every second would crack it in under 3 hours of pure brute force. The only reason your phone PIN is actually secure is the lockout mechanism — phones lock after 5 or 10 failed attempts, escalate the delay, and eventually wipe. Mathematically, the PIN itself is barely a speed bump.

How many possible Rubik's Cube positions are there?

Around 43 quintillion (4.3 × 10^19) — an astronomical number. And yet, any Rubik's Cube position is solvable in at most 20 moves, a result called 'God's number' that was proven by exhaustive computer search in 2010. The cube reveals both the enormity of the search space and, surprisingly, the existence of very short paths through it — a beautiful intersection of combinatorics and computation.

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Permutation & Combination Calculator

Math

Compute permutations (nPr, ordered arrangements) and combinations (nCr, unordered selections) using factorials — ideal for probability, lottery odds, poker hands, and password strength analysis.

Math

Permutation & Combination Calculator

Generated on August 22, 2026

n (total items)

r (chosen)

1 min readHow we build →

Quick Answer

Compute permutations (nPr, ordered arrangements) and combinations (nCr, unordered selections) using factorials — ideal for probability, lottery odds, poker hands, and password strength analysis.

How the Permutation & Combination Calculator works

Math

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!.

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.

By Muhammad Salman Saleem, Full-Stack Web DeveloperMethodologyNo sign-up, no tracking · Calculated on your device, not our servers

What's worth knowing first

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.

Pro tips most people learn the hard way

The order-matters test resolves almost every textbook question. 'Is ABC different from CBA in this problem?' If yes, permutation. If no, combination. Race medals: yes (gold, silver, bronze are distinct). Lottery numbers: no (the ticket is the same set regardless of order). PINs and passwords: yes.
PIN security is mathematical weakness held together by lockout mechanisms. A 4-digit PIN has 10,000 possibilities — crackable in milliseconds without a lockout. The reason your phone is secure isn't the PIN, it's that the OS locks after 5 wrong attempts and escalates from there. Knowing this changes how seriously you should take 4-digit PINs on anything without a lockout.
Combinatorial explosions catch people off guard. 10! is 3.6 million. 20! is 2.4 quintillion. 60! exceeds the number of atoms in the observable universe. This is why exhaustive search ('try every possibility') is fast for small problems and impossible for big ones — the same algorithm that solves 5! in microseconds takes longer than the age of the universe at 30!.
For poker probability calculations, 52C5 = 2,598,960 is the denominator you'll use over and over. Royal flush count: 4. So odds of a royal flush from a random deal are 4 / 2,598,960 = 1 in 649,740. Pair count: 1,098,240, so odds of at least one pair are about 42 percent. The whole probability tree for poker hands hangs off these counts.
Pro tip: when n gets above 20 or so, this calculator's floating-point math starts losing precision. For exact answers on large factorials, you need arbitrary-precision libraries (Python's math.factorial, Java's BigInteger). For 'approximately how many,' Stirling's approximation gives you a quick log-scale estimate without computing the full factorial.

?What is the Permutation & Combination Calculator?

The Formula

nPr = n! / (n − r)!. nCr = n! / [r! × (n − r)!]. nCr = nPr / r!.

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).

nPr and nCr — Common Values Reference Table

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)

Put It in Perspective

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).

Practical Examples

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.

Frequently Asked Questions

Permutations care about order — picking Alice then Bob is different from picking Bob then Alice, because position matters (gold versus silver). Combinations don't care — {Alice, Bob} is the same group whether you wrote down Alice or Bob first. If the problem rewards positions (1st, 2nd, 3rd), it's permutation. If it just cares whether someone is in or out (committee membership, lottery numbers), it's combination.

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

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Calculate logarithms in any base, including common (log10), natural (ln), and binary (log2).

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Premium Converters

Permutation & Combination Calculator

Math

Compute permutations (nPr, ordered arrangements) and combinations (nCr, unordered selections) using factorials — ideal for probability, lottery odds, poker hands, and password strength analysis.

Math

Permutation & Combination Calculator

Generated on August 22, 2026

n (total items)

r (chosen)

1 min readHow we build →

Quick Answer

Compute permutations (nPr, ordered arrangements) and combinations (nCr, unordered selections) using factorials — ideal for probability, lottery odds, poker hands, and password strength analysis.

How the Permutation & Combination Calculator works

Math

Formula

nPr = n! / (n − r)!. nCr = n! / [r! × (n − r)!]. nCr = nPr / r!.

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.

By Muhammad Salman Saleem, Full-Stack Web DeveloperMethodologyNo sign-up, no tracking · Calculated on your device, not our servers

What's worth knowing first

Pro tips most people learn the hard way

The order-matters test resolves almost every textbook question. 'Is ABC different from CBA in this problem?' If yes, permutation. If no, combination. Race medals: yes (gold, silver, bronze are distinct). Lottery numbers: no (the ticket is the same set regardless of order). PINs and passwords: yes.
PIN security is mathematical weakness held together by lockout mechanisms. A 4-digit PIN has 10,000 possibilities — crackable in milliseconds without a lockout. The reason your phone is secure isn't the PIN, it's that the OS locks after 5 wrong attempts and escalates from there. Knowing this changes how seriously you should take 4-digit PINs on anything without a lockout.
Combinatorial explosions catch people off guard. 10! is 3.6 million. 20! is 2.4 quintillion. 60! exceeds the number of atoms in the observable universe. This is why exhaustive search ('try every possibility') is fast for small problems and impossible for big ones — the same algorithm that solves 5! in microseconds takes longer than the age of the universe at 30!.
For poker probability calculations, 52C5 = 2,598,960 is the denominator you'll use over and over. Royal flush count: 4. So odds of a royal flush from a random deal are 4 / 2,598,960 = 1 in 649,740. Pair count: 1,098,240, so odds of at least one pair are about 42 percent. The whole probability tree for poker hands hangs off these counts.
Pro tip: when n gets above 20 or so, this calculator's floating-point math starts losing precision. For exact answers on large factorials, you need arbitrary-precision libraries (Python's math.factorial, Java's BigInteger). For 'approximately how many,' Stirling's approximation gives you a quick log-scale estimate without computing the full factorial.

?What is the Permutation & Combination Calculator?

The Formula

nPr = n! / (n − r)!. nCr = n! / [r! × (n − r)!]. nCr = nPr / r!.

nPr and nCr — Common Values Reference Table

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)

Put It in Perspective

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).

Practical Examples

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.

Frequently Asked Questions

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

Browse all Math

Exponent / Power Calculator

Raise any base to any exponent — positive, negative, fractional, or decimal — to compute powers, reciprocals, and nth roots for compound interest, scientific notation, and computing.

Square Root Calculator

Compute the principal square root, cube root, square, and cube of any real number, with support for negative cube roots and irrational results like √2 for Pythagorean theorem work.

Logarithm Calculator

Calculate logarithms in any base, including common (log10), natural (ln), and binary (log2).

Triangle Solver

Solve any triangle from SSS, SAS, ASA, AAS, SSA, or right-triangle (Pythagoras) inputs. Drawn-to-scale visualization with all sides, angles, area, and perimeter.

3D Solid Volume Calculator

Volume + surface area for sphere, cylinder, cone, cube, and rectangular prism. Interactive 3D visualization — rotate and zoom.

Trigonometry Calculator

Compute sin, cos, tan, csc, sec, cot for any angle in degrees or radians. Interactive unit-circle visualization.

Distance & Midpoint Calculator

Find the distance between two points, the midpoint, slope, and angle. Interactive coordinate-plane visualization.

Statistics Batch Calculator

Mean, median, mode, std deviation, variance, quartiles for any list of numbers. Histogram + box plot visualizations.

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Permutation & Combination Calculator | Premium Converters

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Premium Converters

Permutation & Combination Calculator

Permutation & Combination Calculator

How the Permutation & Combination Calculator works

What's worth knowing first

Pro tips most people learn the hard way

?What is the Permutation & Combination Calculator?

The Formula

nPr and nCr — Common Values Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

Popular Conversions

Embed this calculator on your site

Browse other categories

Premium Converters

Permutation & Combination Calculator

Permutation & Combination Calculator

How the Permutation & Combination Calculator works

What's worth knowing first

Pro tips most people learn the hard way

?What is the Permutation & Combination Calculator?

The Formula

nPr and nCr — Common Values Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

Popular Conversions

Embed this calculator on your site

Browse other categories

Premium Converters

Premium Converters

Permutation & Combination Calculator

Permutation & Combination Calculator

How the Permutation & Combination Calculator works

What's worth knowing first

Pro tips most people learn the hard way

?What is the Permutation & Combination Calculator?

The Formula

nPr and nCr — Common Values Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

What is the difference between a permutation and a combination?

What is a factorial?

When should I use permutation vs combination?

Is nPr always larger than nCr?

What about arrangements with repetition allowed?

What are real-world applications?

Why are 4-digit PINs weak?

How many possible Rubik's Cube positions are there?

Popular Conversions

Embed this calculator on your site

More Math Tools

Browse other categories

Premium Converters

Permutation & Combination Calculator

Permutation & Combination Calculator

How the Permutation & Combination Calculator works

What's worth knowing first

Pro tips most people learn the hard way

?What is the Permutation & Combination Calculator?

The Formula

nPr and nCr — Common Values Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

What is the difference between a permutation and a combination?

What is a factorial?

When should I use permutation vs combination?

Is nPr always larger than nCr?

What about arrangements with repetition allowed?

What are real-world applications?

Why are 4-digit PINs weak?

How many possible Rubik's Cube positions are there?

Popular Conversions

Embed this calculator on your site

More Math Tools

Browse other categories