# Binomial Coefficient Calculator

URL: https://speedycalculator.com/calculators/math/binomial-coefficient/
Last updated: 2026-10-30

> The binomial coefficient calculator computes "n choose k", C(n, k) = n! / (k!(n − k)!), exactly – even for large n, where the result has hundreds of digits – and shows the matching row of Pascal's triangle for small n.

## Inputs

- n (total items)
- k (items chosen)

## Results

- C(n, k)
- Ordered selections P(n, k) = C(n, k) × k!
- Symmetry
- Row n of Pascal's triangle

## Formula

```
C(n, k) = n! ÷ (k! × (n − k)!)
```

## Explanation

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).

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Contemporary Mathematics](https://openstax.org/details/books/contemporary-mathematics)
- [Wolfram MathWorld – Binomial coefficient](https://mathworld.wolfram.com/BinomialCoefficient.html)