Zero Choose Zero

Theorem

$\dbinom 0 0 = 1$

where $\dbinom 0 0$ denotes a binomial coefficient.

Proof 1

By Zero Choose n:

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

where:

$\dbinom 0 n$ denotes a binomial coefficient

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

Hence directly:

$\dbinom 0 0 = 1$

Proof 2

By Binomial Coefficient with Zero:

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

Hence directly:

$\dbinom 0 0 = 1$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $2$