About the Combination Calculator
When you're selecting a group where order doesn't matter - like picking 3 team members from 10 candidates - combinations, not permutations, give the right count. Our Combination Calculator finds nCr, the number of unordered selections.
How It Works
The calculator divides the permutation count by r factorial (the number of ways to reorder the same selected group), since combinations treat every reordering of the same group as identical.
Formula & Methodology
Combinations start from the permutation count and then remove the overcounting caused by treating different orderings of the same group as distinct - since any group of r items can be arranged in r! different orders, dividing the permutation count P(n,r) by r! collapses all those equivalent orderings down to a single count per unique group. That's exactly why C(n,r) = P(n,r) ÷ r! = n! ÷ (r! × (n−r)!).
Step-by-Step: Calculating It By Hand
- 1Confirm r does not exceed n.
- 2Calculate n factorial, r factorial, and (n−r) factorial.
- 3Multiply r! by (n−r)!.
- 4Divide n! by that product to find the number of unordered selections.
Examples
Choosing a team
Selecting an unordered group of 3 people from 10 candidates gives C(10,3) = 120 possible combinations - much smaller than the 720 permutations of the same numbers, since order doesn't matter here.
Lottery-style selection
Choosing 6 numbers from 49 (a common lottery format) gives C(49,6) = 13,983,816 - illustrating just how large combination counts can get even from a modest range.
Advantages
- Correctly calculates unordered selections, distinct from permutations
- Handles large n and r values efficiently
- Useful for probability calculations involving groups or subsets
- Standard tool across statistics, probability, and combinatorics
Common Mistakes
- Using combinations when order actually matters (or vice versa)
- Forgetting combinations are always smaller than or equal to the corresponding permutation count
- Miscounting n or r, especially in multi-step probability problems combining several combination calculations
- Not validating that r doesn't exceed n before calculating
Edge Cases to Watch For
- C(n,r) always equals C(n, n−r) - choosing which r items to include is equivalent to choosing which (n−r) items to exclude.
- When r equals 0 or r equals n, the result is always 1 - there's exactly one way to choose nothing, and exactly one way to choose everything.
- Combination counts are always less than or equal to the corresponding permutation count for the same n and r, since ordering information is discarded.
- This formula assumes no repetition allowed and treats all n items as distinct - combinations with repetition use a different formula.
Common Use Cases
- Calculating unordered group selections for probability problems
- Lottery and card game odds calculations
- Team or committee selection counting problems
- Combinatorics and discrete math coursework