Permutation and Combination Calculator
Compute nPr and nCr — how many ways to arrange or choose r items out of n.
How it works
nPr = n! ÷ (n − r)! · nCr = n! ÷ (r! × (n − r)!)
Permutations count ordered arrangements: n choices for the first slot, n − 1 for the second, and so on for r slots. Combinations count unordered groups, so the permutation count is divided by r! — the number of ways each group can be shuffled. Passwords and podium finishes are permutations; lottery tickets and committees are combinations.
Worked example
From 10 people, how many 3-person outcomes? Ordered (gold, silver, bronze): 10P3 = 10 × 9 × 8 = 720. Unordered (a committee of 3): 10C3 = 720 ÷ 3! = 720 ÷ 6 = 120.
Frequently asked questions
How do I know whether order matters?
Ask if swapping two picks creates a different outcome. Alice-then-Bob differs from Bob-then-Alice in a race result (permutation) but is the same committee (combination).
Why does the calculator stop at n = 170?
Because 170! ≈ 7.3 × 10³⁰⁶ is the largest factorial standard floating-point numbers can hold; 171! overflows. Very large counts are shown in scientific notation.
What if items can repeat?
Different formulas apply: ordered picks with repetition allowed are nʳ, and unordered ones use (n + r − 1)Cr, the stars-and-bars formula. This calculator assumes each item is picked at most once.
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This calculator is for educational purposes. Double-check important results before acting on them.