# Binomial coefficient

The **binomial coefficient** is a part of combinatorics. The binomial coefficient represent the number of possible choices of *k* elements out of *n* labelled elements (with the order of the *k* elements being irrelevant): that is, the number of subsets of size *k* in a set of size *n*. The binomial coefficients are written as ; they are named for their role in the expansion of the binomial expression (*x*+*y*)^{n}.

## Definition

### Example

## Formulae involving binomial coefficients

Specifying a subset of size *k* is equivalent to specifying its complement, a subset of size *n*-*k* and vice versa. Hence

There is just one way to choose *n* elements out of *n* ("all of them") and correspondingly just one way to choose zero element ("none of them").

The number of singletons (single-element sets) is *n*.

The subset of size *k* out of *n* things may be split into those which do not contain the element *n*, which correspond to subset of *n*-1 of size *k*, and those which do contain the element *n*. The latter are uniquely specified by the remaining *k*-1 element which are drawn from the other *n*-1.

There are no subsets of negative size or of size greater than *n*.

### Examples

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## Usage

The binomial coefficient can be used to describe the mathematics of lottery games. For example the German *Lotto* has a system, where you can choose 6 numbers from the numbers 1 to 49. The binomial coefficient is 13,983,816, so the probability to choose the correct six numbers is .

## Binomial coefficients and prime numbers

If *p* is a prime number then *p* divides for every . The converse is also true.