Formulas
- Combinations (order does not matter): C(n, k) = n! ÷ (k! × (n − k)!)
- Permutations of k items (order matters): P(n, k) = n! ÷ (n − k)!
- All items in order: n!
- With repetition: nᵏ when order matters, C(n + k − 1, k) when it does not
Examples
Choosing 3 people out of 10 for a committee: 120 combinations. Choosing a president, a secretary and a treasurer from the same 10: 720 permutations, because the order matters.
A lottery where you pick 6 numbers out of 60 has 50,063,860 possible tickets. A 4-digit PIN with repetition has 10⁴ = 10,000 options.
Combination or permutation?
Ask yourself if swapping two items gives a different result. A pizza with ham and cheese is the same as cheese and ham (combination). A race result of first and second is not (permutation).
Frequently asked questions
What is a factorial?
n! is the product of all whole numbers from 1 to n. 5! = 5 × 4 × 3 × 2 × 1 = 120, and 0! = 1 by definition.
Why does it show “× 10^” for big numbers?
Results with more than 21 digits are shown in scientific notation. 100! has 158 digits.
See also
Results are estimates for information only and are not financial, legal or medical advice. Rates, taxes and fees vary — confirm with a professional or your lender.