Permutation and Combination Calculator
Calculate permutations (where order matters) and combinations (where order doesn't) for any n and r values. Used in probability, lottery math and combinatorics.
How to use the Permutation and Combination Calculator
- Enter your inputs into the Permutation and Combination Calculator above.
- Results update instantly as you type — no submit button needed.
- Adjust any value to see how the result changes in real time.
The permutation and combination formulas
Permutations: nPr = n! / (n−r)! · · · Combinations: nCr = n! / (r!(n−r)!)
Permutations count arrangements (order matters). Combinations count subsets (order doesn't). nCr is always nPr divided by r! (the number of orderings of r items).
Worked example
How many ways to arrange 3 books from a shelf of 7? P(7,3) = 7!/(4!) = 210. How many ways to choose 3 books to take on vacation? C(7,3) = 7!/(3!4!) = 35.
Frequently asked questions
When do I use permutation vs. combination?
Permutation: order matters (election results, race finishes, passwords). Combination: order doesn't matter (lottery picks, sub-committees, hand of cards).
How does this relate to factorials?
Factorial n! = n × (n−1) × ... × 2 × 1 counts all orderings of n items. Permutation and combination formulas are factorial ratios that count specific subsets of those orderings.
Why does C(n, r) = C(n, n−r)?
Choosing r items is equivalent to choosing the n−r to leave out. C(10, 3) = C(10, 7) = 120.