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Permutations & Combinations Calculator
Determine the exact number of ways to select or arrange items from a dataset. Enter your total number of items and how many you are choosing to instantly calculate both permutations and combinations.
How to use this calculator
- Total Items ($n$): Enter the total size of the pool you are choosing from.
- Amount Choosing ($r$): Enter how many items you are selecting from that pool.
- Replacement Mode:
- Without Replacement: Once an item is selected, it is removed from the pool and cannot be chosen again (e.g., dealing a deck of cards or lottery balls).
- With Replacement: Selected items are returned to the pool and can be chosen multiple times (e.g., rolling a die or cracking a passcode).
Permutations & Combinations
Calculate exact arrangements and selections.
Combinations (Order Doesn’t Matter)
0
Formula: n! / (r!(n-r)!)
Permutations (Order Matters)
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Formula: n! / (n-r)!
*Technical note: This calculator uses native BigInt logic. It provides exact, precise integers for massive combinatorics calculations without defaulting to rounded scientific notation. Data inputs are capped at 5,000 to maintain browser performance.
Understanding Your Results
Both permutations and combinations tell you how many possible outcomes exist, but they handle the order of those outcomes differently.
- Combinations ($C$): Use this when order does not matter. For example, if you are choosing a 3-person committee from a 10-person team, selecting Alice, Bob, and Charlie is the exact same committee as Charlie, Bob, and Alice.
- Formula (No replacement): $C(n,r) = \frac{n!}{r!(n-r)!}$
- Permutations ($P$): Use this when order matters. For example, if you are choosing a President, Vice President, and Treasurer, the specific arrangement of Alice, Bob, and Charlie represents a completely different outcome than Charlie, Bob, and Alice.
- Formula (No replacement): $P(n,r) = \frac{n!}{(n-r)!}$