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:

\(\displaystyle \binom n 0\) \(=\) \(\displaystyle \frac {n!} {0! \ n!}\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \frac {n!} {1 \cdot n!}\) Definition of Factorial of $0$
\(\displaystyle \) \(=\) \(\displaystyle 1\)

$\blacksquare$


Also see


Sources