Combinations and permutations calculator
How many ways to arrange or choose r items from n, with and without order mattering.
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Formula
nPr = n! ÷ (n − r)! nCr = n! ÷ (r! × (n − r)!)
How it works
A permutation counts arrangements where order matters; a combination counts selections where it does not. Both start from n! (n factorial), the product of every whole number from 1 to n.
Worked example
The same calculation with the default values:
- Total items (n) 8
- Items chosen (r) 3
- nPr = 8! ÷ (8 − 3)! = 336
- nCr = 8! ÷ (3! × (8 − 3)!) = 56
Result: Permutations (nPr): 336 · Combinations (nCr): 56
Permutations count arrangements where order matters, while combinations count selections where order does not - the same n and r values give a very different (always larger for permutations) result depending on which question is actually being asked.
Common mistakes
- Using the permutation formula when order genuinely doesn't matter (e.g. choosing a committee) or the combination formula when it does (e.g. awarding 1st/2nd/3rd place) - mixing these up gives a result too large or too small by a large factor.
- Entering r larger than n - you cannot choose or arrange more items than exist in the set; this should be flagged as an invalid input, not silently computed.
- Forgetting how quickly factorials grow - n! becomes astronomically large for even moderately sized n, which is why these formulas divide out common factors rather than compute full factorials directly.
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