Permutation & Combination Calculator
Calculate permutations (nPr) and combinations (nCr) for any values of n and r. Free online with formulas and explanations.
Embed this toolPermutation Result
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 |
Advertisement
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.
External References
For more number theory tools, try our GCD & LCM Calculator.