Binomial Coefficient with Self

Theorem

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

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

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


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

Proof
From the definition of binomial coefficient:


 * $\dbinom n n = \dfrac {n!} {n! \ \paren {n - n}!} = \dfrac {n!} {n! \ 0!}$

the result following directly from the definition of the factorial, where $0! = 1$.

From N Choose Negative Number is Zero:
 * $\forall k \in \Z_{<0}: \dbinom n k = 0$

So for $n < 0$:
 * $\dbinom n n = 0$

Also see

 * Particular Values of Binomial Coefficients for other similar results.