Particular Values of Binomial Coefficients

Theorem

$\dbinom 0 0 = 1$

$\dbinom 0 n = \delta_{0 n}$

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

$\dbinom 1 n = \begin{cases} 1 & : n \in \set {0, 1} \\ 0 & : \text {otherwise} \end{cases}$

Let $n \in \Z$ be an integer.

Let $k \in \Z_{<0}$ be a (strictly) negative integer.

Then:

$\dbinom n k = 0$

$\forall r \in \R: \dbinom r 0 = 1$

$\forall r \in \R: \dbinom r 1 = r$

$\forall n \in \Z: \dbinom n n = \sqbrk {n \ge 0}$

where $\sqbrk {n \ge 0}$ denotes Iverson's convention.

That is:

$\forall n \in \Z_{\ge 0}: \dbinom n n = 1$

$\forall n \in \Z_{< 0}: \dbinom n n = 0$

$\forall n \in \N_{>0}: \dbinom n {n - 1} = n$

$\forall r \in \R: \dbinom r 2 = \dfrac {r \paren {r - 1} } 2$