1. Problem 19: Identify the correct statement for the inductive step in mathematical induction. The inductive step shows that if the statement holds for some integer $k$, then it also holds for $k+1$. This is written as: $$P(k) \to P(k+1)$$ This corresponds to option B. 2. Problem 20: Calculate $\frac{10!}{8!}$. Recall the factorial definition: $$n! = n \times (n-1) \times \cdots \times 1$$ So, $$\frac{10!}{8!} = \frac{10 \times 9 \times 8!}{8!}$$ Canceling $8!$: $$\frac{10 \times 9 \times \cancel{8!}}{\cancel{8!}} = 10 \times 9 = 90$$ Therefore, the answer is 90, which corresponds to option B. Final answers: - For question 19: B) $P(k) \to P(k+1)$ - For question 20: B) 90