Permutation and Combination Calculator
Need an accurate permutation and combination calculator with instant factorial breakdowns? This tool functions as a dual nPr and nCr calculator with steps, handling problems with repetition and without repetition across any sample space.
What is the difference between permutation and combination?
The main difference is whether order matters. A permutation counts ordered arrangements where sequence creates a new outcome (1-2-3 is distinct from 3-2-1). A combination counts unordered groups where item position has zero impact on the resulting set.
- Permutations (Order Matters): Assigning distinct roles—like President, Vice President, and Treasurer—is a permutation. Swapping who holds each seat produces a different leadership lineup.
- Combinations (Order Doesn't Matter): Picking three co-equal teammates from a squad of ten is a combination. Picking Alice first or third produces the exact same final roster.
What is the formula for permutation and combination?
Both operations use the factorial (n!) operation—multiplying an integer by every positive whole number beneath it down to 1. Here is how both structures operate across sample sets:
1. Standard Permutation Formula (Without Repetition)
Use the standard nPr formula (n! / (n−r)!) when picking ordered items that cannot be selected twice:
2. Combination Calculator Without Repetition
Use the standard nCr formula (n! / (r!(n−r)!)) when selecting unique items where sequence does not matter:
3. Permutation Calculator With Repetition
When sequence counts and items can repeat continuously (such as multi-digit PINs or character passwords):
4. Combinations With Repetition Allowed
Applied in probability counting problems using the "stars and bars" grouping method:
How to calculate permutations and combinations by hand
Follow this 4-step workflow to evaluate combinations manually without touching an external solver:
Step 1: Identify Total Items (n)Count the entire pool of elements available. For example, a pool of 5 players gives n = 5.
Step 2: Identify Sample Size (r)Identify how many elements you are selecting from the set. Choosing a 3-person lineup sets r = 3.
Step 3: Check Order and Repetition RulesCheck if items can repeat and whether sequence changes identity (Permutation vs Combination).
Substitute values into the formula and simplify terms directly:
Frequently Asked Questions
How do you calculate nCr on a calculator?
On a physical scientific calculator, enter your total items (n), press the nCr button, input your chosen sample size (r), and press enter. To skip key combinations, enter your numbers into our interactive tool above for immediate answers and intermediate steps.
When to use permutation vs combination in probability?
Use permutations when calculating probabilities tied to rankings, passwords, or order-dependent runs within your sample space. Use combinations when finding odds for lottery draws, card hands, or committee selections where order does not matter.
Permutation & Combination Tool
nPr = 10! / (10 - 3)! = 10! / 7! = 720
nCr = 10! / (3! × (10 - 3)!) = 10! / (3! × 7!) = 120
* Permutations (nPr) account for sequential ordering (e.g., race podiums, passcodes), whereas Combinations (nCr) calculate un-ordered subsets (e.g., card hands, team selections). Calculations use arbitrary-precision BigInt arithmetic.
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