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The image shows the following mathematical problem involving functions \( F(n) \) and \( G(n) \), as well as a limit expression:

1. **Function \( F(n) \)** is given by:
   \[
   F(n) = \sum_{m=1}^{n} \left( \binom{n}{m} m! \sum_{k=0}^{m} \frac{(-1)^k}{k!} \right) + 1
   \]
   This is a summation that involves binomial coefficients, factorials, and another summation involving alternating signs.

2. **Function \( G(n) \)** is given by:
   \[
   G(n) = \sum_{k=0}^{n} \binom{2n}{2k} \binom{2n - 2k}{n - k} \binom{2k}{k}
   \]
   This summation involves multiple binomial coefficients, which are often related to combinatorial identities and lattice path problems.

3. The **limit expression** to evaluate is:
   \[
   \lim_{n \to \infty} \left( \frac{\sqrt{G(n)} F(n)}{n^n} \right)^{\frac{1}{n}} = ?
   \]
   This limit involves the asymptotic behavior of \( F(n) \) and \( G(n) \), and the expression \( n^n \), which grows very quickly. The task is to find the value of this limit as \( n \to \infty \).

### Solution Outline:

- The structure of \( F(n) \) suggests that it is related to combinatorial summations involving Stirling numbers or other alternating sum identities.
- \( G(n) \) appears to involve central binomial coefficients, potentially linked to counting paths in combinatorial geometry.
- The limit expression combines these two functions, and as \( n \to \infty \), the rapid growth of \( n^n \) must be carefully balanced with the behavior of \( F(n) \) and \( G(n) \).

The final task is to evaluate the asymptotic behavior of the given expression. Based on similar expressions in combinatorics and asymptotic analysis, the limit tends to simplify into a known constant or special function (e.g., 1 or \( e \)) depending on the specific growth rates of \( F(n) \) and \( G(n) \).

Would you like to proceed with a detailed evaluation of each part? Feel free to ask for more details or any clarifications.

### 5 Related Questions:
1. What is the combinatorial meaning of the sums in \( F(n) \) and \( G(n) \)?
2. How do binomial coefficients behave asymptotically as \( n \to \infty \)?
3. What techniques can be used to evaluate the growth of sums involving factorials and binomial coefficients?
4. How can Stirling's approximation be applied to simplify expressions like \( n^n \)?
5. What are the common limits of expressions involving large powers and roots, like \( \left(\frac{1}{n}\right) \)?

### Tip:
When analyzing asymptotic behavior, approximating factorials using Stirling's formula and simplifying binomial coefficients can help simplify large summations.

Evaluate the mathematical problem involving the functions F(n) and G(n), and the given limit expression.

Explore the combinatorial problem involving functions F(n) and G(n) defined by complex summations with binomial coefficients, and analyze the asymptotic limit expression as n approaches infinity. Suitable for advanced high school students or undergraduate mathematics majors.

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