1. **State the problem:** Simplify the expression $$\frac{9 \cdot 8 \cdot 7 \cdot 6 \cdot 5}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$$ without using a calculator. 2. **Formula and rules:** This is a fraction with products in numerator and denominator. We can simplify by canceling common factors. 3. **Step-by-step simplification:** $$\frac{9 \cdot 8 \cdot 7 \cdot 6 \cdot \cancel{5}}{\cancel{5} \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1}$$ 4. Next, simplify denominator product: $$4 \cdot 3 \cdot 2 \cdot 1 = 24$$ 5. Now rewrite numerator: $$9 \cdot 8 \cdot 7 \cdot 6$$ 6. Simplify numerator and denominator by factoring: Note that $$8 = 4 \cdot 2$$ and denominator is $$24 = 4 \cdot 3 \cdot 2$$. 7. Cancel common factors stepwise: $$\frac{9 \cdot \cancel{8} \cdot 7 \cdot 6}{\cancel{4} \cdot 3 \cdot \cancel{2} \cdot 1} = \frac{9 \cdot (4 \cdot 2) \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2}$$ Canceling 4 and 2: $$\frac{9 \cdot \cancel{4} \cdot \cancel{2} \cdot 7 \cdot 6}{\cancel{4} \cdot 3 \cdot \cancel{2}} = \frac{9 \cdot 7 \cdot 6}{3}$$ 8. Simplify $$\frac{9}{3} = \cancel{3} \cdot 3 / \cancel{3} = 3$$: $$\frac{\cancel{9} \cdot 7 \cdot 6}{\cancel{3}} = 3 \cdot 7 \cdot 6$$ 9. Multiply remaining numbers: $$3 \cdot 7 = 21$$ $$21 \cdot 6 = 126$$ 10. **Final answer:** $$\boxed{126}$$