Particular Values of Binomial Coefficients

Theorem

Binomial Coefficient $\dbinom 0 0$

$\dbinom 0 0 = 1$

Binomial Coefficient $\dbinom 0 n$

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

Binomial Coefficient $\dbinom 1 n$

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

N Choose Negative Number is Zero

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

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

Then:

$\dbinom n k = 0$

Binomial Coefficient with Zero

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

Binomial Coefficient with One

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

Binomial Coefficient with Self

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

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

That is:

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

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

Binomial Coefficient with Self minus One

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

Binomial Coefficient with Two

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

Also see

• Properties of Binomial Coefficients