Definition:Binomial Coefficient

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Definition

Definition 1

Let $n \in \Z_{\ge 0}$ and $k \in \Z$.

Then the binomial coefficient $\dbinom n k$ is defined as:

$\dbinom n k = \begin{cases} \dfrac {n!} {k! \paren {n - k}!} & : 0 \le k \le n \\ & \\ 0 & : \text { otherwise } \end{cases}$

where $n!$ denotes the factorial of $n$.


Definition 2

Let $n \in \Z_{\ge 0}$ and $k \in \Z$.

The number of different ways $k$ objects can be chosen (irrespective of order) from a set of $n$ objects is denoted:

$\dbinom n k$

This number $\dbinom n k$ is known as a binomial coefficient.


Definition 3

Let $n \in \Z_{\ge 0}$ and $k \in \Z$.

Then the binomial coefficient $\dbinom n k$ is defined as the coefficient of the term $a^k b^{n - k}$ in the expansion of $\left({a + b}\right)^n$


Definition for Real Numbers

Let $r \in \R, k \in \Z$.

Then $\dbinom r k$ is defined as:

$\dbinom r k = \begin{cases} \dfrac {r^{\underline k}} {k!} & : k \ge 0 \\ & \\ 0 & : k < 0 \end{cases}$

where $r^{\underline k}$ denotes the falling factorial.


That is, when $k \ge 0$:

$\displaystyle \binom r k = \frac {r \left({r - 1}\right) \cdots \left({r - k + 1}\right)} {k \left({k - 1}\right) \cdots 1} = \prod_{j \mathop = 1}^k \frac {r + 1 - j} j$

It can be seen that this agrees with the definition for integers when $r$ is an integer.


For most applications the integer form is sufficient.


Definition for Complex Numbers

Let $z, w \in \C$.

Then $\dbinom z w$ is defined as:

$\dbinom z w := \displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}$

where $\Gamma$ denotes the Gamma function.


When $z$ is a negative integer and $w$ is not an integer, $\dbinom z w$ is infinite.


Definition for Multiindices

Let $k = \left \langle {k_j}\right \rangle_{j \mathop \in J}$ and $\ell = \left \langle {\ell_j}\right \rangle_{j \mathop \in J}$ be multiindices.

Let $\ell \le k$.


Then $\dbinom k \ell$ is defined as:

$\displaystyle \binom k \ell = \prod_{j \mathop \in J} \binom {k_j} {\ell_j}$


Note that since by definition only finitely many of the $k_j$ are non-zero, the product in the definition of $\dbinom k \ell$ is convergent.


Notation

The notation $\dbinom n k$ for the binomial coefficient was introduced by Andreas Freiherr von Ettingshausen in his $1826$ work Die kombinatorische Analysis, als Vorbereitungslehre zum Studium der theoretischen höheren Mathematik.

It appears to have become the de facto standard in recent years.

As a result, $\dbinom n k$ is frequently voiced the binomial coefficient $n$ over $k$.


Other notations include:

$C \left({n, k}\right)$
${}^n C_k$
${}_n C_k$
$C^n_k$
${C_n}^k$

all of which can cause a certain degree of confusion.


Examples

$2$ from $5$

The number of ways of choosing $2$ objects from a set of $5$ is:

$\dbinom 5 2 = \dfrac {5 \times 4} {2 \times 1} = \dfrac {5!} {2! \, 3!} = 10$


$3$ from $7$

The number of ways of choosing $3$ objects from a set of $7$ is:

$\dbinom 7 3 = \dfrac {7 \times 6 \times 5} {3 \times 2 \times 1} = \dfrac {7!} {3! \, 4!} = 35$


$4$ from $52$

The number of ways of choosing $4$ objects from a set of $52$ (for example, cards from a deck) is:

$\dbinom {52} 4 = \dfrac {52 \times 51 \times 50 \times 49} {4 \times 3 \times 2 \times 1} = \dfrac {52!} {48! \, 4!} = 270 \, 725$


Number of Bridge Hands

The total number $N$ of possible different hands for a game of bridge is:

$N = \dfrac {52!} {13! \, 39!} = 635 \ 013 \ 559 \ 600$


Historical Note

The binomial coefficients have been known about since at least the ancient Greeks and Romans, who were familiar with them for small values of $k$.

See the historical note to Pascal's Triangle for further history.


Also see

  • Results about binomial coefficients can be found here.


Technical Note

The $\LaTeX$ code to render the binomial coefficient $\dbinom n k$ can be written in the following ways:

\dbinom n k

or:

\displaystyle {n \choose k}


The \dbinom form is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$ because it is simpler.

It is in fact an abbreviated form of \displaystyle \binom n k, which is the preferred construction when \displaystyle is required for another entity in the expression.

While the form \binom n k is valid $\LaTeX$ syntax, it renders the entity in the reduced size inline style: $\binom n k$ which $\mathsf{Pr} \infty \mathsf{fWiki}$ does not endorse.


To render compound arguments, braces are needed to delimit the parameter when using \dbinom, but (confusingly) not \choose.

For example, to render $\dbinom {a + b} {c d}$ the following can be used:

\dbinom {a + b} {c d}

or:

\displaystyle {a + b \choose c d} $\displaystyle {a + b \choose c d}$

Again, for consistency across $\mathsf{Pr} \infty \mathsf{fWiki}$, the \dbinom form is preferred.