Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.5 The Binomial Theorem - 12.5 Assess Your Understanding - Page 836: 12

$1$

Recall: (1) ${n\choose j}=\dfrac{n!}{(n-j)!j!}$. (2) $0!=1$ Hence, \begin{align*} \require{cancel} {1000\choose 0}&=\dfrac{1000!}{(1000-0)!0!}\\ &=\dfrac{1000!}{1000!0!}\\ &=\dfrac{1000!}{1000!}\\ &=1\end{align*}