Definition:Binomial Coefficient/Integers/Definition 1
Definition
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 $n$.
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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Binomial Coefficients: $3.1.2$: Binomial Coefficients
- 1964: A.M. Yaglom and I.M. Yaglom: Challenging Mathematical Problems With Elementary Solutions: Volume $\text { I }$ ... (previous) ... (next): Problems
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.6$. Algebra of congruences: Example $42 \ (2)$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 19$: Combinatorial Analysis
- 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 3$: The Binomial Formula and Binomial Coefficients: $3.5$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.18$: Sequences Defined Inductively: Exercise $3$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 3$: Natural Numbers: Exercise $\S 3.11 \ (4)$
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.2$ The Binomial Theorem
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $21$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: Exercise $12$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $24$
- 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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(2)$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $24$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): binomial coefficients
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): binomial coefficients