Combinations & Permutations Calculator
Compute the number of combinations (nCr) and permutations (nPr) for choosing r items from n, along with the factorials — the building blocks of counting and probability.
- n!
- 3,628,800
- r!
- 6
Combinations vs. permutations
Both count the ways to choose r items from n, but they differ on whether order matters. Permutations count ordered arrangements (so ABC and CBA are different); combinations count unordered selections (ABC and CBA are the same).
- Use permutations when order matters — e.g. arranging people in a line or ranking finishers.
- Use combinations when order doesn’t matter — e.g. dealing a hand of cards or choosing a committee.
Formulas
Permutations (order matters):
Combinations (order doesn’t matter):
Worked examples
- A 5-card poker hand from 52 cards: C(52, 5) = 2,598,960 combinations (order doesn’t matter).
- Choosing a president, vice-president, and treasurer from 10 people: P(10, 3) = 720 permutations (roles are ordered).
- A 6-number lottery from 49: C(49, 6) = 13,983,816 combinations.
Combinations and permutations in Excel
Use =COMBIN(n, r) for combinations and =PERMUT(n, r) for permutations. =FACT(n) gives the factorial n!.
Frequently asked questions
What is the difference between a combination and a permutation?
A permutation counts ordered arrangements, so different orders are counted separately. A combination counts unordered selections, so the same items in a different order count only once. Permutations are always greater than or equal to combinations for the same n and r.
What is nCr?
nCr, written C(n, r) or “n choose r”, is the number of ways to choose r items from n when order does not matter. It equals n! / (r!(n − r)!).
How many combinations are there in a lottery?
For a lottery where you pick 6 numbers from 49, there are C(49, 6) = 13,983,816 possible combinations.