Binomial coefficient (n choose 0) equals 1

Theorem

The following formula holds: $${n \choose 0} = 1,$$ where ${n \choose 0}$ denotes the binomial coefficient.

Proof

From the definition, $${n \choose k} = \dfrac{n!}{k! (n-k)!},$$ so for $k=0$ we get, using the fact that $0!=1$, $${n \choose 0} = \dfrac{n!}{0! (n-0)!} = \dfrac{n!}{n!} = 1,$$ as was to be shown.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $3.1.5$