Zero Choose n

Theorem

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

where:

$\dbinom 0 n$ denotes a binomial coefficient

$\delta_{0 n}$ denotes the Kronecker delta.

Proof

By definition of binomial coefficient:

$\dbinom m n = \begin{cases}

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

Thus when $n > 0$:

$\dbinom 0 n = 0$

and when $n = 0$:

$\dbinom 0 0 = \dfrac {0!} {0! \paren {0 - 0}!} = 1$

by definition of factorial.

Hence the result by definition of the Kronecker delta.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(48)$