Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.5 The Binomial Theorem - 11.5 Assess Your Understanding - Page 857: 11

$1$

According to the binomial theorem, we have: $\displaystyle{n\choose k}=\dfrac{n!}{(n-k)! \ k!}$. Since $0!=1$ Substitute $1000$ for $n$ and $1000$ for $k$ in the above formula. Therefore, $\dbinom{1000}{1000}=\dfrac{1000!}{1000! \ (1000-1000)!} \\ =\dfrac{1000!}{0! \ 1000!}\\=\dfrac{1000 !}{1000!}\\= 1$