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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 \( n \).

Simplify the expression: \( \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} \div \left( \frac{1}{6!} - \frac{1}{7!} + \frac{1}{8!} \right) \).

Learn how to simplify the complex factorial expression involving factorials from 5! to 8!. This step-by-step guide shows how to simplify and divide fractions with factorials, suitable for students in Grades 9-12.

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