To show that \( n! \in \Omega(2^n) \), we need to demonstrate that there exists a positive constant \( c \) and a positive integer \( n_0 \) such that for all \( n \geq n_0 \):

\[
n! \geq c \cdot 2^n
\]

### Step 1: Consider the Ratio \(\frac{n!}{2^n}\)

We start by considering the ratio of \( n! \) to \( 2^n \):

\[
\frac{n!}{2^n} = \frac{1 \cdot 2 \cdot 3 \cdot \ldots \cdot n}{2^n}
\]

This can be rewritten as:

\[
\frac{n!}{2^n} = \frac{1}{2} \cdot \frac{2}{2} \cdot \frac{3}{2} \cdot \ldots \cdot \frac{n}{2}
\]

\[
= \prod_{k=1}^{n} \frac{k}{2}
\]

### Step 2: Simplify and Estimate the Product

Notice that for \( k \geq 2 \), \( \frac{k}{2} \geq 1 \), so:

\[
\frac{n!}{2^n} = \prod_{k=1}^{n} \frac{k}{2} \geq \frac{n}{2} \cdot 1 \cdot 1 \cdot \ldots \cdot 1 = \frac{n}{2}
\]

Thus:

\[
\frac{n!}{2^n} \geq \frac{n}{2}
\]

This inequality shows that as \( n \) increases, \( \frac{n!}{2^n} \) grows larger. For sufficiently large \( n \), this ratio is greater than any constant, meaning \( n! \) grows faster than \( 2^n \).

### Step 3: Conclusion

Since we have shown that:

\[
\frac{n!}{2^n} \geq \frac{n}{2} \quad \text{for all sufficiently large } n
\]

We can conclude that:

\[
n! \geq c \cdot 2^n \quad \text{for some constant } c > 0
\]

Thus, \( n! \in \Omega(2^n) \).

---

Would you like to see more details or have any questions? Here are 5 related questions:

1. How does \( n! \) compare to \( n^n \) in terms of asymptotic growth?
2. What is the significance of the \( \Omega \)-notation in algorithm analysis?
3. Can you show that \( n! \) also belongs to \( \theta(n^n) \)?
4. What are the differences between \( \Omega \)-notation and \( O \)-notation?
5. How can Stirling's approximation be used to analyze the growth of \( n! \)?

**Tip:** When working with factorials, it's often useful to compare their growth with exponential or polynomial functions to determine asymptotic bounds.

Learn how to prove that n factorial (n!) belongs to Omega of 2 raised to the power of n (Ω(2^n)) with a detailed mathematical proof. Explore the ratio analysis and asymptotic growth comparisons used in this proof. Suitable for undergraduate students and those interested in asymptotic notation in algorithm analysis.

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