We are tasked with evaluating:

$\frac{7! \times 5!}{6! \times 8!}.$

Step 1: Break down the factorial expressions.

We know the following factorial values:

• $7! = 7 \times 6!$
• $8! = 8 \times 7!$

So we can rewrite the expression as:

$\frac{7! \times 5!}{6! \times 8!} = \frac{7 \times 6! \times 5!}{6! \times 8 \times 7!}.$

Step 2: Cancel out common terms.

The $6!$ in both the numerator and the denominator cancel out:

$\frac{7 \times 5!}{8 \times 7!}.$

Now, rewrite $7!$ as $7 \times 6!$:

$\frac{7 \times 5!}{8 \times 7 \times 6!}.$

The $7$ in the numerator and denominator cancel out:

$\frac{5!}{8 \times 6!}.$

Step 3: Evaluate the remaining factorials.

Now calculate $5! = 120$ and $6! = 720$:

$\frac{120}{8 \times 720}.$

Simplify the denominator:

$\frac{120}{5760}.$

Step 4: Final simplification.

Now divide:

$\frac{120}{5760} = \frac{1}{48}.$

Final Answer:

$\boxed{\frac{1}{48}}.$

Would you like more details or have any questions about this?

Here are 5 related questions:

1. What is the value of $9!$?
2. How do you simplify expressions involving factorials?
3. What is the result of $\frac{5! \times 6!}{7!}$?
4. How can we evaluate factorials for larger numbers efficiently?
5. How does the factorial operation relate to combinatorics?

Tip: Cancel common terms early when working with factorials to simplify calculations.