Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.5 The Binomial Theorem - 12.5 Assess Your Understanding - Page 836: 16

$17,672,631,900$

Recall: ${n\choose j}=\dfrac{n!}{(n-j)!j!}$. Hence, using the formula above and a calculator gives \begin{align*} \require{cancel} {37\choose 19}&=\dfrac{37!}{(37-19)!19!}\\ &=\dfrac{37!}{18!19!}\\ &=17,672,631,900\end{align*}