Permutation (nPr) & Combination (nCr) Calculator
Use our free online Permutation & Combination Calculator to perform fast, accurate calculations with instant formulas, step-by-step arithmetic, and verified reference benchmarks.
Permutation & Combination Calculator
Instant Live Computation • Zero Waiting
Adjust inputs on the left for instantaneous live computation.
How to Calculate: The Permutation & Combination Calculator Formula
Permutations count ordered arrangements where position matters. Combinations count unordered selections where grouping order is irrelevant.
Step-by-Step Calculation Guide
- Input total population pool size (n).
- Input number of items chosen in sample subset (r).
- Review Permutations (nPr) and Combinations (nCr).
- Inspect factorial expansion.
Practical Permutation & Combination Calculator Examples
Pick 5 lottery numbers from a pool of 50 balls.
8 runners competing for Gold, Silver, Bronze (1st, 2nd, 3rd).
Permutations vs Combinations Comparison Matrix
Official reference values and benchmark classifications based on standard institutional guidelines.
| Property | Permutations (nPr) | Combinations (nCr) |
|---|---|---|
| Does Order Matter? | YES (Order is distinct) | NO (Order is arbitrary) |
| Mathematical Formula | n! / (n - r)! | n! / [ r!(n - r)! ] |
| Relative Output Size | Always larger (multiplied by r!) | Always smaller |
| Classic Real-World Example | Lock combinations, race ranking | Poker hands, lottery numbers |
Frequently Asked Questions about Permutation & Combination Calculator
Why is a 'combination lock' misnamed?
A combination lock is mathematically a permutation lock because entering the correct numbers in the wrong order will not open the lock.
What is 0! (zero factorial)?
By mathematical definition and convention, 0! = 1, ensuring combinatorics formulas evaluate consistently when r = n or r = 0.
How are combinations used in poker?
A standard 52-card deck yields 52C5 = 2,598,960 possible unique 5-card poker hands, used to establish game hand rankings.
What is Pascal's Triangle?
Pascal's Triangle is a geometric arrangement where the nth row and rth column gives the exact combination value nCr.