The expression you're asked to evaluate is:

$\frac{6!}{8!}$

Step-by-step solution:

1. First, recall the definition of a factorial. For any integer $n$, the factorial $n!$ is the product of all positive integers from $1$ to $n$.

   • $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
   • $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320$

2. Now, simplify the fraction: $\frac{6!}{8!} = \frac{6!}{8 \times 7 \times 6!}$

3. Notice that $6!$ appears in both the numerator and denominator, so they cancel out: $\frac{6!}{8 \times 7 \times 6!} = \frac{1}{8 \times 7}$

4. Finally, simplify the product in the denominator: $8 \times 7 = 56$

So, the simplified answer is: $\frac{1}{56}$

Would you like further details or have any questions?

Here are five related questions to explore:

1. What is the factorial of 10, and how can it be used in calculations?
2. How do you simplify fractions involving factorials in more complex expressions?
3. Can you evaluate the expression $\frac{7!}{9!}$ following the same process?
4. How would you handle the expression $\frac{n!}{(n+2)!}$ for general $n$?
5. How are factorials used in combinatorics and probability?

Tip: When dealing with factorials in fractions, always look for common terms to cancel out, which makes the simplification much easier.