A factorial, written n!, is the product of all positive integers from 1 to n. So 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1, which keeps formulas like nCr consistent when r = 0 or r = n. Factorials grow extremely fast — 10! is already 3.6 million.

When should I use permutation vs combination?

Ask yourself: does the order of the chosen items matter? If yes (race results, passwords, arrangements), it is a permutation. If no (committees, lottery numbers, card hands), it is a combination. The question to ask is always 'is ABC different from CBA in this problem?'.

Is nPr always larger than nCr?

Yes, for r > 1, because nPr = nCr × r!. Each combination of r items can be arranged in r! different orders, so permutations count each combination r! times. For r = 0 and r = 1, they are equal.

What about arrangements with repetition allowed?

The formulas change. Permutations with repetition (pick from n, with replacement, in r positions) = n^r. Combinations with repetition (pick r from n, with replacement, order doesn't matter) = (n + r − 1) C r. For example, the number of ways to choose 3 scoops of ice cream from 10 flavors if repeats are allowed is (10 + 3 − 1) C 3 = 220.

What are real-world applications?

Probability (card games, lottery, dice), password and PIN strength analysis, cryptography, experimental design (how many variants can we test?), bioinformatics (protein sequencing), quality sampling, and tournament scheduling. Almost any problem involving counting possibilities uses these formulas.

Why are 4-digit PINs weak?

Only 10,000 possibilities (10^4, permutation with repetition). A phone trying one PIN per second would crack it in less than 3 hours. That is why phones lock or erase after repeated failures — the PIN itself is mathematically weak; only the lockout mechanism keeps it secure.

How many possible Rubik's Cube positions are there?

Approximately 43 quintillion (4.3 × 10^19). Despite that enormous number, any Rubik's Cube can be solved in at most 20 moves — the 'God's number' proven by computer search in 2010. Combinatorics reveals both the scale of the search space and, sometimes, surprisingly short paths through it.

Permutation & Combination Calculator

Math

Calculate nPr (arrangements) and nCr (selections) for combinatorial problems.

Math

Permutation & Combination Calculator

Generated on April 24, 2026

n (total items)

r (chosen)

?What is the Permutation & Combination Calculator?

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.

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).

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

A permutation cares about order — picking Alice then Bob is different from picking Bob then Alice. A combination does not — {Alice, Bob} is the same regardless of order. If the problem rewards positions (1st, 2nd, 3rd), use permutation. If it only cares about membership (chosen or not chosen), use combination.

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Permutation & Combination Calculator

Permutation & Combination Calculator

?What is the Permutation & Combination Calculator?

The Formula

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?

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