1. The problem asks which is NOT a step in mathematical induction. 2. Mathematical induction typically involves these steps: - Base case: Prove $P(1)$ is true. - Inductive hypothesis: Assume $P(k)$ is true for some arbitrary $k$. - Inductive step: Prove $P(k+1)$ is true assuming $P(k)$ is true. 3. Checking the options: - A) Prove $P(1)$: This is the base case, a valid step. - B) Assume $P(k)$: This is the inductive hypothesis, a valid step. - D) Prove $P(k+1)$ under assumption $P(k)$: This is the inductive step, a valid step. - C) Prove $P(k-1)$: This is NOT a step in induction because induction moves forward from $k$ to $k+1$, not backward. 4. Therefore, the answer to question 15 is option C. 5. The second problem asks: If $n! = 120$, find $n$. 6. Recall factorial definition: $n! = n \times (n-1) \times \cdots \times 1$. 7. Calculate factorials: - $4! = 4 \times 3 \times 2 \times 1 = 24$ - $5! = 5 \times 4! = 5 \times 24 = 120$ - $6! = 6 \times 5! = 6 \times 120 = 720$ 8. Since $5! = 120$, the value of $n$ is 5. 9. Therefore, the answer to question 16 is option B. Final answers: - 15: C - 16: B