Permutation & Combination Calculator
Permutations (nPr) and combinations (nCr) of n items taken r.
Input sheet
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Counting how many ways things can be arranged or chosen is the heart of probability, lottery odds, password strength, and scheduling. Permutations count ordered selections; combinations count unordered ones.
How it works
nPr = n! ÷ (n−r)! counts ordered selections; nCr = nPr ÷ r! counts unordered ones.
Use a permutation (nPr) when order matters — a race podium, a PIN, a seating order. Use a combination (nCr) when it does not — a hand of cards, a committee, a pizza topping set. Every combination corresponds to r! permutations, which is why nCr = nPr ÷ r!.
This tool computes both from n (total items) and r (how many you pick), and also reports n! for context.
nPr = n! ÷ (n−r)!; nCr = nPr ÷ r! = n! ÷ (r! × (n−r)!)
Worked examples
Choosing 3 from 10 items (n=10, r=3). → 120 combinations, 720 permutations
nPr = 10 × 9 × 8 = 720; nCr = 720 ÷ 3! = 720 ÷ 6 = 120.
Picking 2 from 5 (n=5, r=2). → 10 combinations, 20 permutations
nPr = 5 × 4 = 20; nCr = 20 ÷ 2! = 20 ÷ 2 = 10.
Tips & gotchas
- Ask 'does order matter?' first. If rearranging the same items counts as different, use permutations; if not, use combinations.
- nCr is symmetric: choosing r from n equals choosing (n−r) from n. nC3 from 10 equals nC7 from 10.
- Permutations are always at least as large as combinations for the same n and r, since they distinguish orderings.
FAQ
When do I use a permutation versus a combination?
Permutations when the arrangement order matters (codes, rankings, sequences); combinations when only the chosen set matters (teams, hands, samples).
Why is nCr smaller than nPr?
Combinations ignore order, so each unordered set is counted once instead of r! times. Dividing the permutation count by r! gives the combination count.
What does nC0 or nCn equal?
Both equal 1. There is exactly one way to choose nothing and exactly one way to choose everything.
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