1. **State the problem:** We want to approximate the factorial of 75, denoted as $75!$, using Stirling's approximation. 2. **Formula:** Stirling's approximation for $n!$ is given by: $$n! \approx \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$$ where $e$ is Euler's number, approximately 2.71828. 3. **Apply the formula for $n=75$:** $$75! \approx \sqrt{2 \pi \times 75} \left(\frac{75}{e}\right)^{75}$$ 4. **Calculate each part:** - Calculate $\sqrt{2 \pi \times 75}$: $$\sqrt{2 \times 3.1416 \times 75} = \sqrt{471.2389} \approx 21.7156$$ - Calculate $\left(\frac{75}{e}\right)^{75}$: $$\left(\frac{75}{2.71828}\right)^{75} = (27.605)^{75}$$ 5. **Combine the parts:** $$75! \approx 21.7156 \times (27.605)^{75}$$ 6. **Interpretation:** The exact value of $75!$ is extremely large, so this approximation gives a very close estimate. **Final approximate value:** $$75! \approx 2.4809 \times 10^{109}$$ This means $75!$ is approximately $2.48$ followed by $109$ zeros. This completes the approximation using Stirling's formula.