Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.5 The Binomial Theorem - 12.5 Assess Your Understanding - Page 836: 6

$35$

Recall: ${n\choose j}=\dfrac{n!}{(n-j)!j!}$. Hence, \begin{align*} \require{cancel} {7\choose 3}&=\dfrac{7!}{(7-3)!3!}\\ &=\dfrac{7!}{4!3!}\\ &=\dfrac{7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{4\cdot3\cdot2\cdot1\cdot3\cdot2\cdot1}\\ &=\dfrac{7\cdot6\cdot5\cdot4\cdot\cancel{3\cdot2\cdot1}}{4\cdot3\cdot2\cdot1\cdot\cancel{3\cdot2\cdot1}}\\ &=\dfrac{840}{24}\\ &=35\end{align*}