The binomial coefficient counts how many ways you can choose k items from n when order does not matter: the number of possible lottery tickets, poker hands or committees. It is also the coefficient of xᵏ in (1 + x)ⁿ, which gives it its name.
How to use the Binomial Coefficient Calculator
- n (total items)
- k (items chosen)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The calculator multiplies and divides step by step with exact whole numbers (BigInt) in your browser, so there is no rounding even for C(1000, 500).
Formula
C(n, k) = n! ÷ (k! × (n − k)!)
Worked example
Inputs — n (total items): 52; k (items chosen): 5.
Results — C(n, k) = 2,598,960; Ordered selections P(n, k) = C(n, k) × k! = 311,875,200; Symmetry = C(52, 5) = C(52, 47).
Key terms
- Combination
- A selection where order does not matter.
- Factorial n!
- The product 1 × 2 × … × n.
Frequently asked questions
How many 5-card poker hands are there?
C(52, 5) = 2,598,960.
What is C(n, 0)?
1 – there is exactly one way to choose nothing.
Why is C(n, k) = C(n, n − k)?
Choosing k items to keep is the same as choosing n − k items to leave out.
How is this related to Pascal's triangle?
Row n of the triangle lists C(n, 0) … C(n, n), and each entry is the sum of the two above it.