Our Permutation and Combination Calculator lets you quickly compute permutations (nPr) and combinations (nCr), with or without repetition. It is designed for students, educators, researchers, and professionals who need fast, reliable results and clear formulas.
What the Permutation and Combination Calculator Does
The calculator evaluates two fundamental counting methods in combinatorics: permutations and combinations. Use permutations when order matters, and combinations when order does not matter. You can also toggle whether repetition is allowed, switching seamlessly between classic formulas and their repetition-inclusive counterparts.
- Permutation (nPr): Counts ordered arrangements without repetition.
- Combination (nCr): Counts unordered selections without repetition.
- Permutation with repetition: Uses n^r, where items can be reused and order matters.
- Combination with repetition: Uses C(n + r - 1, r), where items can repeat and order does not matter.
Key Formulas
Without Repetition
- Permutation: nPr = n! / (n ? r)!
- Combination: nCr = n! / (r! (n ? r)!)
With Repetition
- Permutation with repetition: n^r
- Combination with repetition: C(n + r ? 1, r)
When to Use Each Option
Choose permutations when the sequence or order of selected items matters. For example, arranging finalists on a podium is a permutation problem. Choose combinations when order does not matter, such as selecting a committee from a group.
Allow repetition if the same item can be chosen multiple times. A lock code where digits can repeat is a permutation with repetition. Selecting scoops of ice cream where flavors can repeat (e.g., two scoops of vanilla) is a combination with repetition.
How to Use the Calculator
- Select the calculation type: Permutation (nPr) or Combination (nCr).
- Choose whether repetition is allowed.
- Enter total items n and selection size r as non-negative integers.
- Click Calculate to see the result. Optionally, show steps for insight into the formula used.
Practical Examples
Permutation without Repetition
If you have n = 10 runners and want to know the number of ways to award gold, silver, and bronze (r = 3), use 10P3 = 10 × 9 × 8 = 720.
Combination without Repetition
From a class of n = 12 students, how many ways can you choose a team of r = 4? Use 12C4 = 495, since the order of selection does not matter.
Permutation with Repetition
How many r = 4-digit codes can you make from n = 10 digits (0–9) if digits can repeat? Answer: 10^4 = 10,000.
Combination with Repetition
How many ways to choose r = 3 scoops from n = 5 flavors if flavors can repeat? Answer: C(5 + 3 ? 1, 3) = C(7, 3) = 35.
Why This Calculator Is Useful
Beyond producing fast results, this tool clarifies which formula fits your scenario. The step-by-step option shows the intermediate setup so you can learn how the final number is derived, reinforcing understanding for homework, exams, or professional analysis.
Tips and Common Pitfalls
- Check whether order matters before deciding between permutations and combinations.
- Confirm whether repetition is allowed; it drastically changes the count.
- For calculations without repetition, ensure 0 ? r ? n; otherwise, the count is zero or invalid.
- Start with smaller values to validate your approach before scaling up.
Use the Permutation and Combination Calculator whenever you need quick, accurate counts for arrangements and selections, with clear formulas and optional steps to support learning and verification.