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Yes, the statement is **true**.

This is a Big-O notation analysis of the factorial function \( n! \), and it aligns with **Stirling's approximation**, which estimates the growth of \( n! \) as \( n \) becomes large. The approximation of \( n! \) is:

\[
n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n
\]

For asymptotic behavior, we focus on the dominant term \( \left( \frac{n}{e} \right)^n \). Thus, the factorial \( n! \) grows at a rate very close to this expression, with additional terms being lower order.

The inequality:

\[
n! \leq C \cdot \left( \frac{n}{e} \right)^n
\]

is an expression of the Big-O notation, where \( C \) is some constant, and \( O\left( \left( \frac{n}{e} \right)^n \right) \) means that \( n! \) does not grow faster than some constant multiple of \( \left( \frac{n}{e} \right)^n \).

Would you like more details, or do you have any questions?

Here are 5 related questions that expand on this topic:

1. How does Stirling's approximation help in simplifying factorial calculations?
2. What is the significance of Big-O notation in algorithmic analysis?
3. Can you derive the constant \( C \) in the Big-O bound of \( n! \)?
4. How do we handle the lower-order terms in Stirling's approximation?
5. What are some practical applications of asymptotic approximations like \( n! \) in computer science?

**Tip:** Big-O notation describes the upper bound of the growth rate of a function, making it particularly useful for analyzing algorithms' worst-case performance.

Explore the Big-O notation analysis of the factorial function n!, showing that n! = O((n/e)^n). This involves Stirling's approximation, which is used to estimate the growth rate of n! for large n. Suitable for students studying asymptotic analysis and algorithmic complexity at the college level.

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