binomial coefficient
For integers $n\ge r\ge 0$ we define
$$\left(\genfrac{}{}{0pt}{}{n}{r}\right)=\frac{n!}{(nr)!r!}$$ 
and call such numbers binomial coefficients^{}.
Properties.

1.
$\left(\genfrac{}{}{0pt}{}{n}{r}\right)$ is an integer (proof. (http://planetmath.org/NchooseRIsAnInteger)).

2.
$\left(\genfrac{}{}{0pt}{}{n}{r}\right)=\left(\genfrac{}{}{0pt}{}{n}{nr}\right)$.

3.
$\left(\genfrac{}{}{0pt}{}{n}{r1}\right)+\left(\genfrac{}{}{0pt}{}{n}{r}\right)=\left(\genfrac{}{}{0pt}{}{n+1}{r}\right)$ (Pascal’s rule).

4.
$\left(\genfrac{}{}{0pt}{}{n}{0}\right)=1=\left(\genfrac{}{}{0pt}{}{n}{n}\right)$ for all $n$.

5.
$\left(\genfrac{}{}{0pt}{}{n}{0}\right)+\left(\genfrac{}{}{0pt}{}{n}{1}\right)+\left(\genfrac{}{}{0pt}{}{n}{2}\right)+\mathrm{\cdots}+\left(\genfrac{}{}{0pt}{}{n}{n}\right)={2}^{n}$.

6.
$\left(\genfrac{}{}{0pt}{}{n}{0}\right)\left(\genfrac{}{}{0pt}{}{n}{1}\right)+\left(\genfrac{}{}{0pt}{}{n}{2}\right)\mathrm{\cdots}+{(1)}^{n}\left(\genfrac{}{}{0pt}{}{n}{n}\right)=0$ for $n>0$.

7.
${\sum}_{t=k}^{n}\left(\genfrac{}{}{0pt}{}{t}{k}\right)=\left(\genfrac{}{}{0pt}{}{n+1}{k+1}\right)$.
Properties 5 and 6 are the binomial theorem^{} applied to ${(1+1)}^{n}$ and ${(11)}^{n}$, respectively, although they also have purely combinatorial meaning.
Motivation
Suppose $n\ge r$ are integers. The below list shows some examples where the binomial coefficients appear.

•
$\left(\genfrac{}{}{0pt}{}{n}{r}\right)$ constitute the coefficients when expanding the binomial ${(x+y)}^{n}$ – hence the name binomial coefficients. See Binomial Theorem.

•
$\left(\genfrac{}{}{0pt}{}{n}{r}\right)$ is the number of ways to choose $r$ objects from a set with $n$ elements.

•
On the context of computer science, it also helps to see $\left(\genfrac{}{}{0pt}{}{n}{r}\right)$ as the number of strings consisting of ones and zeros with $r$ ones and $nr$ zeros. This equivalency comes from the fact that if $S$ be a finite set^{} with $n$ elements, $\left(\genfrac{}{}{0pt}{}{n}{r}\right)$ is the number of distinct subsets of $S$ with $r$ elements. For each subset $T$ of $S$, consider the function
$${X}_{T}:S\to \{0,1\}$$ where ${X}_{T}(x)=1$ whenever $x\in T$ and $0$ otherwise (so ${X}_{T}$ is the characteristic function^{} for $T$). For each $T\in \mathcal{P}(S)$, ${X}_{T}$ can be used to produce a unique bit string of length $n$ with exactly $r$ ones.
Notes
The $\left(\genfrac{}{}{0pt}{}{n}{r}\right)$ notation was first introduced by von Ettinghausen [1] in 1826, altough these numbers have been used long before that. See this page (http://planetmath.org/PascalsTriangle) for some notes on their history. Although the standard mathematical notation for the binomial coefficients is $\left(\genfrac{}{}{0pt}{}{n}{r}\right)$, there are also several variants. Especially in high school environments one encounters also $C(n,r)$ or ${C}_{r}^{n}$ for $\left(\genfrac{}{}{0pt}{}{n}{r}\right)$.
Remark. It is sometimes convenient to set $\left(\genfrac{}{}{0pt}{}{n}{r}\right):=0$ when $r>n$. For example, property 7 above can be restated: ${\sum}_{t=1}^{n}\left(\genfrac{}{}{0pt}{}{t}{k}\right)=\left(\genfrac{}{}{0pt}{}{n+1}{k+1}\right)$. It can be shown that $\left(\genfrac{}{}{0pt}{}{n}{r}\right)$ is elementary recursive.
References
 1 N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.
Title  binomial coefficient 
Canonical name  BinomialCoefficient 
Date of creation  20130322 11:47:25 
Last modified on  20130322 11:47:25 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  32 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 11B65 
Classification  msc 05A10 
Classification  msc 19D55 
Classification  msc 19K33 
Classification  msc 19D10 
Synonym  combinations^{} 
Synonym  choose 
Related topic  PascalsRule 
Related topic  BinomialTheorem 
Related topic  BernoulliDistribution2 
Related topic  MultinomialTheorem 
Related topic  ProofOfLucassTheorem2 
Related topic  Factorial^{} 
Related topic  CentralBinomialCoefficient 
Related topic  PascalsTriangle 
Related topic  TaylorSeriesViaDivision 
Related topic  CombinationsWithRepeatedElements 
Related topic  NonIsomorphicGroupsOfGivenOrder 
Related topic  AppellSeque 