Definition:Binomial Coefficient/Integers/Definition 3
Definition
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 $\paren {a + b}^n$.
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.
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:
\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}
or:
\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.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man"
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: Exercise $12$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): binomial coefficient: 1.
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients