Permutation & Combination Calculator
Calculate nPr, nCr, and combinations with repetition (nHr), with the full factorial work shown step by step.
Find out how many ways you can choose r items from a set of n. This calculator computes permutations (nPr), combinations (nCr), and combinations with repetition (nHr) all at once, and walks through each factorial expansion so you can see exactly how the formula produces the answer.
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How to use
- Enter the total number of items (n) and how many you're choosing (r) in their respective fields.
- Results for permutations, combinations, and combinations with repetition update live as you type.
- Expand the step-by-step section below to study each factorial expansion and how the values are substituted into the formula.
- Copy any result to drop straight into a report or your notes.
FAQ
- What's the key difference between nPr and nCr?
- Whether order matters. Permutations treat a different arrangement as a different outcome, giving n!/(n-r)!. Combinations only care about which items you picked, not the order, so you divide by an extra r! to remove the duplicate arrangements.
- When would I use combinations with repetition (nHr)?
- Any time you're picking from a set and can pick the same option more than once — like counting the possible vote tallies when 5 anonymous ballots are cast among 3 candidates. It follows the formula (n+r-1)Cr.
- What happens if the number to choose (r) is larger than the total number of items (n)?
- Permutations (nPr) and combinations (nCr) can't select more items than actually exist, so when r is greater than n, the result comes out to 0. Homogeneous combinations (nHr) are the exception — since repeated selections are allowed, they still produce a valid, nonzero result even when r exceeds n.