1. **Problem Statement:** Calculate the factorial of a very large number, such as $1000!$. 2. **Definition:** The factorial of a positive integer $n$, denoted $n!$, is the product of all positive integers from 1 to $n$: $$n! = 1 \times 2 \times 3 \times \cdots \times n$$ 3. **Challenge:** Directly computing $1000!$ by multiplying all numbers is impractical due to its enormous size. 4. **Approach:** Use Stirling's approximation, a formula that estimates factorials for large $n$: $$n! \approx \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$$ where $e$ is Euler's number (approximately 2.71828). 5. **Explanation:** This approximation gives a very close estimate of $n!$ without computing every multiplication. 6. **Example:** For $1000!$, Stirling's formula is: $$1000! \approx \sqrt{2 \pi \times 1000} \left(\frac{1000}{e}\right)^{1000}$$ 7. **Practical Use:** For exact values, use software or libraries that support arbitrary-precision arithmetic (e.g., Python's math.factorial or specialized libraries). 8. **Summary:** For very large factorials, use Stirling's approximation for estimation or computational tools for exact values.