Permutation & Combination Calculator
Calculate permutations (nPr) and combinations (nCr) for any values of n and r. Free online with formulas and explanations.
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Quick Reference Table
| n | r | nPr | nCr |
|---|---|---|---|
| 5 | 3 | 60 | 10 |
| 8 | 2 | 56 | 28 |
| 10 | 4 | 5,040 | 210 |
| 6 | 6 | 720 | 1 |
| 12 | 5 | 95,040 | 792 |
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Understanding Permutations & Combinations
Permutations and combinations are fundamental concepts in combinatorics, the branch of mathematics that deals with counting. Both answer the question "How many ways can I select r items from a set of n items?" — but they differ in whether order matters.
Permutation Formula (nPr)
When the order of selection matters, use the permutation formula:
P(n,r) = n! / (n − r)!For example, the number of ways to award 1st, 2nd, and 3rd place among 5 runners is P(5,3) = 5! / 2! = 60.
Combination Formula (nCr)
When the order does not matter, use the combination formula:
C(n,r) = n! / (r! × (n − r)!)For example, the number of ways to choose a team of 3 people from a group of 5 is C(5,3) = 5! / (3! × 2!) = 10.
Key Differences
- Order matters: Permutations count ordered arrangements; combinations count unordered subsets.
- Relationship: C(n,r) = P(n,r) / r! — each combination of r items can be arranged in r! ways.
- Use cases: Passwords, rankings, and sequences use permutations. Committees, lottery numbers, and groups use combinations.
Frequently Asked Questions
Permutations count arrangements where order matters. Combinations count selections where order does not matter. For example, choosing a president and vice president from a group is a permutation because the roles are distinct. Choosing a committee of two people is a combination because the order in which they are selected does not matter.