Evaluate.
!7!5!/6!8
Simplify your answer as much as possible. 


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.

Evaluate. !7!5!/6!8 Simplify your answer as much as possible.

This guide simplifies the factorial expression 7! × 5! / 6! × 8! using key mathematical principles. Follow the step-by-step breakdown, including the use of factorial properties and fraction simplification, ideal for high school students (Grades 9-12).

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