Combinations and permutations calculator
Combinations count ways to choose k items from n when order doesn't matter. Permutations count arrangements when order does matter.
Combinations C(n,k)
—
Order doesn't matter
Permutations P(n,k)
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Order matters
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Formulas
Combinations: C(n, k) = n! / (k! × (n − k)!)
Permutations: P(n, k) = n! / (n − k)!
Examples
| n | k | C(n,k) | P(n,k) |
|---|---|---|---|
| 5 | 2 | 10 | 20 |
| 10 | 3 | 120 | 720 |
| 52 | 5 | 2,598,960 | 311,875,200 |
The 52-choose-5 example is the number of 5-card poker hands from a standard deck (combinations) versus the number of distinct ordered 5-card deals (permutations).
References
- Combination (binomial coefficient n!/(k!(n−k)!))Wikipedia · en.wikipedia.org
- Permutation (ordered selections, n!/(n−k)!)Wikipedia · en.wikipedia.org
Frequently asked
What is the difference between combinations and permutations?
Combinations: the order of selection doesn't matter (choosing a committee of 3 from 10 people). Permutations: the order matters (awarding gold, silver, and bronze to 3 from 10 people). C(10,3) = 120; P(10,3) = 720.
What is the formula for combinations?
C(n,k) = n! / (k! × (n−k)!). This equals P(n,k) / k! because combinations divide out the k! ways to arrange the chosen items.
What is the formula for permutations?
P(n,k) = n! / (n−k)!. For k = n, this is simply n!.
What does C(n,0) equal?
C(n,0) = 1 for any n ≥ 0 — there is exactly one way to choose nothing.
How do I share my calculation?
Click "Share with my numbers" to copy a URL that saves your n and k values.
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