Combination Calculator
Determine how many ways you can select or arrange items from a larger set. Enter n and r to calculate combinations, permutations, and supporting factorials.
Combinations C(n, r)
120
Unordered selections
Permutations P(n, r)
720
Ordered arrangements
Factorial n!
3,628,800
Useful reference value
| Quantity | Value | Interpretation |
|---|---|---|
| n! | 3,628,800 | Ways to arrange all n items |
| r! | 6 | Ways to arrange the selected items |
| (n − r)! | 5,040 | Ways to arrange the unselected items |
How to Use This Calculator
- Enter the total number of distinct items and how many you want to choose.
- Review the computed combination count for unordered selections.
- Check the permutation result if order matters in your scenario.
- Use the factorial values to understand the building blocks behind the formulas.
Formula
C(n, r) = n! / (r! · (n − r)!)
P(n, r) = n! / (n − r)!
Both formulas assume selections without replacement. For combinations with repetition, different formulas apply.
Full Description
Combinatorics quantifies how many ways there are to pick or arrange items. Combinations ignore order, while permutations treat distinct orderings as different outcomes. These counts appear in probability, statistics, and game design to enumerate possibilities.
This calculator supports up to 500 items, providing exact integer results using iterative multiplication to avoid overflow. It reports factorial components so you can double-check intermediate values.
Frequently Asked Questions
What if r is larger than n?
Selecting more items than available is impossible without replacement, so the combination count is zero. The calculator highlights this scenario so you can adjust your inputs.
How are large factorials handled?
Factorials grow extremely quickly. The calculator uses double-precision arithmetic, so very large n can exceed numeric limits. Within the 500-item cap, results remain interpretable for most use cases.
Do these formulas assume replacement?
No. They assume each item can be chosen at most once. For selection with replacement, combinations with repetition use a different formula C(n + r − 1, r).
How does this relate to Pascal's triangle?
Binomial coefficients C(n, r) appear in Pascal's triangle. Each entry is the sum of the two above it, which offers an intuitive way to compute combinations iteratively.