Permutation and Combination Calculator – nPr and nCr
Calculate nPr and nCr from whole-number n and r, and learn when order matters. Inputs are restricted to 0≤r≤n≤170.
Permutations count ordered selections; combinations count selections where order is irrelevant. Both use the same n and r inputs, but answer different questions. This tool returns both counts so you can see the difference for one example.
How it works
nPr = n! ÷ (n−r)!. nCr = n! ÷ (r! × (n−r)!). The calculator uses a multiplicative loop and accepts integers with 0 ≤ r ≤ n ≤ 170.
Example
Choose two people from five. If first and second positions matter, 5P2 = 5 × 4 = 20 arrangements. If order does not matter, 5C2 = 10 groups.
How to use
- Enter n, the number of available distinct objects, as a whole number from 0 to 170.
- Enter r, the number selected, as a whole number between 0 and n.
- Read both permutation (ordered) and combination (unordered) counts and choose the one that matches your question.
Key features
- Calculates permutation and combination counts together
- Validates integer n and r ranges
- Highlights the order-versus-no-order difference
Common uses
- Solve introductory probability exercises
- Count possible team selections
- Count rankings or seating selections
Limitations to know
Counts assume distinct selectable objects with no replacement. Large results may exceed the precision of ordinary floating-point numeric calculations; the 170 limit is not a promise of exact integer precision for every huge count.
Frequently asked questions
When do I use nPr?
Use nPr when different orders count as different outcomes, such as awarding first and second place.
When do I use nCr?
Use nCr when the same selected group counts once regardless of its internal order.
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