Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.5 The Binomial Theorem - 12.5 Assess Your Understanding - Page 836: 8

$36$

Recall: ${n\choose j}=\dfrac{n!}{(n-j)!j!}$. Hence, \begin{align*} \require{cancel} {9\choose 7}&=\dfrac{9!}{(9-7)!7!}\\ &=\dfrac{9!}{2!7!}\\ &=\dfrac{9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{2\cdot1\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}\\ &=\dfrac{9\cdot8\cancel{\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}}{2\cdot1\cancel{\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}}\\ &=\dfrac{72}{2}\\ &=36\end{align*}