Zero Choose n
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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: $(48)$