Permutation and Combination Calculator
Calculate permutations P(n,r) and combinations C(n,r) instantly. Find ordered arrangements and unordered selections for any n and r.
Permutation P(10, 3) — order matters
720
10! / (10 − 3)! = 10! / 7!
Combination C(10, 3) — order doesn't matter
120
10! / (3! × 7!)
Relationship: P(10, 3) = C(10, 3) × 3! → 120 × 3! = 720
Reference Table — n = 10, r = 0 to 10
| r | P(n, r) | C(n, r) |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 10 | 10 |
| 2 | 90 | 45 |
| 3 | 720 | 120 |
| 4 | 5,040 | 210 |
| 5 | 30,240 | 252 |
| 6 | 151,200 | 210 |
| 7 | 604,800 | 120 |
| 8 | 1,814,400 | 45 |
| 9 | 3,628,800 | 10 |
| 10 | 3,628,800 | 1 |
What Are Permutations and Combinations?
Permutations and combinations are fundamental concepts in combinatorics, the branch of mathematics dealing with counting. A permutation (nPr) counts the number of ways to arrange r items chosen from n, where order matters. A combination (nCr) counts the number of ways to select r items from n, where order does not matter. These formulas are widely used in probability, statistics, cryptography, and everyday problems like lottery odds and seating arrangements.
Formulas
- Permutation: P(n, r) = n! / (n − r)! — ordered arrangements of r items from n
- Combination: C(n, r) = n! / (r! × (n − r)!) — unordered selections of r items from n
- Relationship: P(n, r) = C(n, r) × r! — every combination has r! ordered permutations
How to Use
- n — total number of items in the set
- r — number of items to choose or arrange
- Results show nPr, nCr, and a step-by-step formula breakdown
- Use the reference table to explore results for different r values at a fixed n
FAQ
When should I use permutation vs combination?
Use permutation when order matters — e.g., how many ways can 3 runners finish 1st, 2nd, and 3rd from a group of 10? Use combination when order does not matter — e.g., how many ways can you choose 3 team members from 10 people?
What is 0! (zero factorial)?
By convention, 0! = 1. This ensures that C(n, 0) = 1 (there is exactly one way to choose nothing) and C(n, n) = 1 (there is exactly one way to choose all items).
What is the difference between C(n, r) and C(n, n−r)?
They are always equal: C(n, r) = C(n, n−r). Choosing r items to include is equivalent to choosing (n−r) items to exclude. For example, C(10, 3) = C(10, 7) = 120.
What are combinations with repetition?
Standard combinations assume each item can only be chosen once. Combinations with repetition (multiset combinations) allow repeated choices and use the formula C(n + r − 1, r). This applies to problems like choosing r scoops of ice cream from n flavors when repeats are allowed.