Combination Calculator
Calculate combinations C(n,k) and permutations P(n,k) with or without repetition. Our 4-in-1 calculator provides all results with step-by-step formulas.
Combinatorics Inputs
Use non-repetition formulas when order matters (permutations) or does not (combinations). If you can choose items multiple times, the calculator also handles repetition scenarios.
Step-by-step reasoning appears below once you calculate.
How to Use This Calculator
Enter the total pool size (n) and the number of selections (k). Click calculate to reveal permutations and combinations for both repetition modes. Reset restores the example pair (n=10, k=3).
Methodology
The calculator uses factorial-based formulas. Non-repetition results compute P(n, k) = n! / (n - k)! and C(n, k) = n! / [k! (n - k)!]. Repetition formulas are nk for permutations and C(n + k - 1, k) for combinations with repeats.
Decision Matrix
| Scenario | Order Matters? | Repetition Allowed? | Formula Type |
|---|---|---|---|
| Selecting a podium finish (1st, 2nd, 3rd) | Yes (Permutation) | No | Permutation P(n, k) |
| Selecting 3 friends for a party | No (Combination) | No | Combination C(n, k) |
| Locker code (digits can repeat) | Yes (Permutation) | Yes | nk |
| Choosing donuts (multiple of the same type) | No (Combination) | Yes | C(n + k - 1, k) |
Key Formulas (No Repetition)
Frequently Asked Questions (FAQ)
What is the formula for a combination without repetition?
The formula is: C(n, k) = n! / [k!(n - k)!]. This counts selections where order does not matter.
When is a result a permutation?
Permutations apply when the arrangement order matters, such as passwords or podium finishes.
Can combinations have repetition?
Yes. When duplicates are allowed (choose donuts with the same flavor), use C(n + k - 1, k).