Binomial Coefficient with Zero
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Theorem
- $\forall r \in \R: \dbinom r 0 = 1$
The usual presentation of this result is:
- $\forall n \in \N: \dbinom n 0 = 1$
Proof
From the definition of binomial coefficients:
- $\dbinom r k = \dfrac {r^{\underline k} } {k!}$ for $k \ge 0$
where $r^{\underline k}$ is the falling factorial.
In turn:
- $\ds x^{\underline k} := \prod_{j \mathop = 0}^{k - 1} \paren {x - j}$
But when $k = 0$, we have:
- $\ds \prod_{j \mathop = 0}^{-1} \paren {x - j} = 1$
as $\ds \prod_{j \mathop = 0}^{-1} \paren {x - j}$ is a vacuous product.
From the definition of the factorial we have that $0! = 1$.
Thus:
- $\forall r \in \R: \dbinom r 0 = 1$
$\blacksquare$
Integer Coefficients
This is completely compatible with the result for natural numbers:
- $\forall n \in \N: \dbinom n 0 = 1$
From the definition:
\(\ds \binom n 0\) | \(=\) | \(\ds \frac {n!} {0! \ n!}\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n!} {1 \cdot n!}\) | Definition of Factorial of $0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
Also see
- Particular Values of Binomial Coefficients for other similar results.
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.2$ The Binomial Theorem
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(4)$