Definition:Binomial Coefficient/Integers/Definition 1

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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 $n$.


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:

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

all of which can cause a certain degree of confusion.

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


\ds {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 \ds \binom n k, which is the preferred construction when \ds 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}


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

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