Binomial Coefficient with Zero/Integer Coefficients
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Theorem
- $\forall n \in \N: \dbinom n 0 = 1$
where $\dbinom n 0$ denotes a binomial coefficient.
Proof
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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Binomial Coefficients: $3.1.5$
- 1964: A.M. Yaglom and I.M. Yaglom: Challenging Mathematical Problems With Elementary Solutions: Volume $\text { I }$ ... (previous) ... (next): Problems
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 19$: Combinatorial Analysis: Theorem $19.10$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.18$: Sequences Defined Inductively: Exercise $3 \ \text{(a)}$
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients: $1.27$