Permutation and Combination Calculator

How it works

Use permutations when order matters, like first, second and third place. Use combinations when it doesn’t, like choosing a team. A combination is the permutation count divided by r!.

Example

How many ways can you pick 3 people from 12?

Combinations 220 · Permutations 1,320

Steps

1. 1
   Order matters (permutations): P(12, 3) = 12 × 11 × 10 = 1,320.
2. 2
   Order doesn’t matter (combinations): divide by 3! = 6, the number of ways to arrange 3 items.
3. 3
   C(12, 3) = 1,320 ÷ 6 = 220.

More examples

In how many ways can 10 runners take 1st, 2nd and 3rd?

Combinations 120 · Permutations 720

Steps

1. 1
   Order matters (permutations): P(10, 3) = 10 × 9 × 8 = 720.
2. 2
   Order doesn’t matter (combinations): divide by 3! = 6, the number of ways to arrange 3 items.
3. 3
   C(10, 3) = 720 ÷ 6 = 120.

How many 5-card hands come from a 52-card deck?

Combinations 2,598,960 · Permutations 311,875,200

Steps

1. 1
   Order matters (permutations): P(52, 5) = 52 × 51 × 50 × 49 × 48 = 311,875,200.
2. 2
   Order doesn’t matter (combinations): divide by 5! = 120, the number of ways to arrange 5 items.
3. 3
   C(52, 5) = 311,875,200 ÷ 120 = 2,598,960.

Common mistake

Ask whether swapping two picks gives a different result. If it doesn’t, it’s a combination.

Questions

What does n! mean?

n factorial: n × (n − 1) × … × 1. So 5! = 120, and 0! is 1.

What if repeats are allowed?

This calculator counts picks without repeats. With repeats, multiply the choices: a 4-digit PIN has 10⁴ = 10,000 options.

Why are there more permutations than combinations?

Each group of r items can be put in r! orders. Picking 3 of 10 gives 720 permutations, and 720 ÷ 3! = 120 combinations.

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