1. Problem statement: Compute $5!$. 2. Formula and rule: For integer $n\ge 0$ the factorial is defined by $$n! = n \times (n-1) \times \dots \times 2 \times 1$$. 3. Important note: By definition $0! = 1$ and factorial multiplies all positive integers down to 1. 4. Expand for $5$: $$5! = 5 \times 4 \times 3 \times 2 \times 1$$. 5. Compute stepwise: $5 \times 4 = 20$. 6. Continue: $20 \times 3 = 60$. 7. Continue: $60 \times 2 = 120$. 8. Multiply by 1: $120 \times 1 = 120$. 9. Final answer: $5! = 120$.