The given expression simplifies to:

$\frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} \div \left( \frac{1}{6!} - \frac{1}{7!} + \frac{1}{8!} \right) = \frac{400}{49}$

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

Here are some related questions:

1. What is the general formula for a factorial?
2. How do factorials grow as the number increases?
3. Can the value of $\frac{400}{49}$ be further simplified?
4. How do you handle more complex factorial expressions?
5. What are some practical applications of factorials in probability and statistics?

Tip: Factorials grow extremely fast, making numbers very large even for small $n$.