Permutation and Combination Calculator
Choose n and r
Results
| Permutations (Order Matters, No Repetition) — P(n, r) | — |
|---|---|
| Combinations (Order Doesn't Matter, No Repetition) — C(n, r) | — |
| Permutations with Repetition — n^r | — |
| Combinations with Repetition — C(n+r-1, r) | — |
How to Use This Permutation and Combination Calculator
Counting problems look easy on the surface—"choose three from ten"—until you realize order might matter, repetition might be allowed, and factorials blow up fast on a basic calculator. This page takes n (total items) and r (how many you pick) and returns all four standard counts in one table, plus step-by-step factorial math when you need to show work.
- Enter n—the size of your set (positive integer, max 10,000).
- Enter r—how many items you choose or arrange (non-negative integer, r ≤ n for no-repetition rows).
- Press Calculate. Four results appear: permutations, combinations, permutations with repetition, and combinations with repetition.
- Read the step block. Factorial expansions and intermediate products print below when the numbers are small enough to display clearly.
- Connect to probability. Combination counts feed binomial problems—pair this with the Probability Calculator when you move from counting to likelihood.
Permutation and Combination Formulas and Practical Applications
Ask one question first: does rearranging the same picks count as a different outcome? If yes, you are in permutation territory; if no, combinations. If you can reuse the same item, add "with repetition" to the label.
Permutations without repetition
P(n, r) = n! / (n − r)!
Arranging r distinct items from n when order matters—race finishes, locker codes without repeated digits, seating charts.
Combinations without repetition
C(n, r) = n! / (r! × (n − r)!)
Choosing r items from n when order does not matter—lottery balls, pizza toppings, survey samples.
With repetition
P with repetition = n^r
C with repetition = C(n + r − 1, r)
PIN codes and dice rolls use n^r. Choosing r items from n types when you can pick the same type again—think scoops of ice cream from n flavors.
Standard Units and Conversion Tables
Example counts for n = 5, r = 2:
| Type | Formula | Result |
|---|---|---|
| P(n,r) no rep. | 5!/3! | 20 |
| C(n,r) no rep. | 5!/(2!×3!) | 10 |
| P with rep. | 5² | 25 |
| C with rep. | C(6,2) | 15 |
Browse all math calculators for statistics and probability helpers.
Frequently Asked Questions
What is the difference between permutations and combinations?
Permutations care about order; combinations do not. AB and BA count as two permutations but one combination.
What does n and r mean in this calculator?
n is the pool size; r is how many you draw. Standard no-repetition formulas require 0 ≤ r ≤ n.
When do I use permutations with repetition?
When order matters and items can repeat—n^r covers cases like a three-digit code from ten digits with repeats allowed.
What is the maximum n or r allowed?
Each caps at 10,000. Larger factorials use big-number math so the page stays stable instead of crashing your browser.
How does this relate to probability?
Binomial probability uses C(n,k) to count success arrangements. After counting, switch to the Probability Calculator for P(X = k) and related totals.
Disclaimer. RapidRatio is informational only—not academic or professional advice. Verify counting assumptions with your instructor before exam day.