Combination, Permutation and Factorial Calculator

Permutation or combination?

Both answer the question "in how many ways" and the difference is whether order matters. Use a permutation when it does and a combination when it does not. Choosing a chair, a deputy and a treasurer from 10 people is a permutation: P(10, 3) = 10 · 9 · 8 = 720. Choosing a committee of 3 from the same 10 people is a combination: C(10, 3) = 720 / 6 = 120.

Formulas

• Factorial: n! = n · (n − 1) · … · 2 · 1 and 0! = 1.
• Permutation: P(n, r) = n! / (n − r)!
• Combination: C(n, r) = n! / (r! · (n − r)!)
• Repeated items: when some of the n objects are identical, n! / (n₁! · n₂! · …)
• Combination with repetition: choosing r items from n kinds when a kind may be picked again, C(n + r − 1, r)
• Circular permutation: n people can sit at a round table in (n − 1)! ways.

Examples

The number of 5 card hands from a 52 card deck is C(52, 5) = 2 598 960. The letters of MISSISSIPPI can be arranged in 11! / (4! · 4! · 2!) = 34 650 different ways, because I and S appear four times each and P twice. Five people can sit at a round table in 4! = 24 ways.

Why is the result so long?

Factorials grow very quickly: 20! has nineteen digits and 100! has 158. Pocket calculators round such numbers. This tool uses whole number arithmetic and gives every digit, together with the digit count and scientific notation.

Limits

n can be at most 10 000. n and r must be whole numbers, and factorials of fractional values (the gamma function) are outside the scope of this tool. It counts arrangements only and does not compute probabilities.