What is the difference between a permutation and a combination?

In a permutation, order matters (ABC ≠ BAC). In a combination, order does not (ABC = BAC). P(n,r) = r! × C(n,r).

When do I use combinations?

When selecting items without caring about their arrangement: lottery numbers, hand of cards, committee selection.

What does 'with repetition' mean?

Items can be chosen more than once. With repetition: C = (n+r−1)!/(r!(n-1)!), P = nʳ.

Permutation and Combination Calculator

Enter n (total items) and r (selected items) to compute P(n,r) for ordered selections and C(n,r) for unordered selections, plus repetition variants and n! factorial.

How to use this calculator

Enter n (total items in the pool) and r (items to select).
Read C(n,r) for combinations (order does not matter) and P(n,r) for permutations (order matters).
Repetition variants and the factorial n! are also shown for reference.

Formula & method

Permutation and combination formulas

C(n,r) = n! / (r! × (n−r)!) · P(n,r) = n! / (n−r)! · C with rep = (n+r−1)! / (r! × (n−1)!) · P with rep = nʳ

n! = n × (n−1) × … × 1. Limited to n ≤ 170 to avoid floating-point overflow.

Example

n=10, r=3: C(10,3) = 10! / (3! × 7!) = 120. P(10,3) = 10! / 7! = 720.

Key insights

Combinations C(n,r): choosing 3 pizza toppings from 10 = C(10,3) = 120.
Permutations P(n,r): arranging 3 runners on a podium from 10 = P(10,3) = 720.
P(n,r) = r! × C(n,r) — permutations are combinations multiplied by the number of arrangements.
With repetition: combinations = C(n+r−1, r), permutations = nʳ.

How to interpret your result

C(n,r) — Combination

The number of ways to choose r items from n without regard to order. Used in probability, lottery, and statistics.

P(n,r) — Permutation

The number of ways to arrange r items from n where order matters. Used in scheduling, race placements, and password generation.

Frequently asked questions

What is the difference between a permutation and a combination?: In a permutation, order matters (ABC ≠ BAC). In a combination, order does not (ABC = BAC). P(n,r) = r! × C(n,r).
When do I use combinations?: When selecting items without caring about their arrangement: lottery numbers, hand of cards, committee selection.
What does 'with repetition' mean?: Items can be chosen more than once. With repetition: C = (n+r−1)!/(r!(n-1)!), P = nʳ.

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