Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.5 The Binomial Theorem - 12.5 Assess Your Understanding - Page 836: 5

$10$

Recall: ${n\choose j}=\dfrac{n!}{(n-j)!j!}$. Hence, \begin{align*} \require{cancel} {5\choose 3}&=\dfrac{5!}{(5-3)!3!}\\ &=\dfrac{5!}{2!3!}\\ &=\dfrac{5\cdot4\cdot3\cdot2\cdot1}{2\cdot1\cdot3\cdot2\cdot1}\\ &=\dfrac{5\cdot4\cdot\cancel{3\cdot2\cdot1}}{2\cdot1\cdot\cancel{3\cdot2\cdot1}}\\ &=\dfrac{20}{2}\\ &=10\end{align*}