Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.5 The Binomial Theorem - 12.5 Assess Your Understanding - Page 836: 7

$21$

Recall: ${n\choose j}=\dfrac{n!}{(n-j)!j!}$. Hence, \begin{align*} \require{cancel} {7\choose 5}&=\dfrac{7!}{(7-5)!5!}\\ &=\dfrac{7!}{2!5!}\\ &=\dfrac{7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{2\cdot1\cdot5\cdot4\cdot3\cdot2\cdot1}\\ &=\dfrac{7\cdot6\cancel{\cdot5\cdot4\cdot3\cdot2\cdot1}}{2\cdot1\cdot\cancel{5\cdot4\cdot3\cdot2\cdot1}}\\ &=\dfrac{42}{2}\\ &=21\end{align*}