About this Permutation & Combination
In probability and statistics, we often need to count how many ways we can select items from a group. The key difference is order. If the order of selection matters (like a lock code), it is a Permutation. If order doesn't matter (like a lottery ticket), it is a Combination.
Who this calculator is for
- Students learning counting principles for probability and statistics.
- Anyone comparing order-matters versus order-does-not-matter scenarios.
- People solving selection and arrangement problems for cards, teams, rankings, or passwords.
When to use it
- Use it before a probability problem when you need to count possible arrangements or selections.
- Use it to check whether a scenario calls for nPr or nCr.
- Use it when you need fast combinatorics values without calculating factorials manually.
Key assumptions
- Inputs should be whole numbers with 0 <= r <= n.
- This calculator assumes you are choosing from distinct items without replacement.
- Permutation counts orderings, while combination removes duplicate orderings of the same chosen set.
Accuracy and testing
Review formula sources, default test cases, assumptions, unit handling, rounding rules, and changelog for this calculator.
How this calculator works
The math relies on factorials ($n!$):
* Permutation (nPr): $\frac{n!}{(n-r)!}$. Counts arrangements where order matters (ABC is different from CBA).
* Combination (nCr): $\frac{n!}{r!(n-r)!}$. Counts selections where order is irrelevant (ABC is the same as CBA).
Inputs explained
n (Total Items)
The total size of the pool you are choosing from.
r (Chosen Items)
The number of items you are selecting or arranging.
Understanding your results
Permutations will always be larger (or equal) to Combinations because every unique ordering counts as a separate outcome. Combinations are smaller because re-ordered duplicates are removed.
Example scenarios
Race Medals
10 runners (n), top 3 get Gold, Silver, Bronze. Order matters. Use Permutation (10P3) = 720 ways.
Card Hand
52 cards (n), deal 5 (r). The order you hold them doesn't matter. Use Combination (52C5) = 2,598,960 hands.
Key Features
- Instant Calculation
- Private & Secure
- Mobile Optimized
Tips & Best Practices
- Mnemonic: 'My fruit salad is a **combination** of apples, grapes, and bananas' (order doesn't matter).
- Mnemonic: 'The **combination** to the safe is 4-7-2' (order matters, so technically it's a **permutation** lock!).
Glossary of Terms
- Factorial (!)
- The product of an integer and all the integers below it (e.g., 4! = 4×3×2×1 = 24).
- nPr
- Notation for Permutations of n items taken r at a time.
- nCr
- Notation for Combinations of n items taken r at a time.