Equivalence of Definitions of Binomial Coefficient

Theorem

The following definitions of the concept of Binomial Coefficient are equivalent:

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 $n$.

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.

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 $\paren {a + b}^n$.

$\Box$

This is proved in the Binomial Theorem.

$\blacksquare$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 19$: Combinatorial Analysis