https://file.aiquickdraw.com/mathbot/file/173323680555238uakas7.jpg

To solve the problem, let us analyze \( \frac{6! \cdot 4!}{5! \cdot 7!} \).

### Step 1: Factorials
- \( 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \)
- \( 4! = 4 \cdot 3 \cdot 2 \cdot 1 \)
- \( 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \)
- \( 7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \)

### Step 2: Simplification
Write the expression in expanded form:
\[
\frac{6! \cdot 4!}{5! \cdot 7!} = \frac{(6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) \cdot (4 \cdot 3 \cdot 2 \cdot 1)}{(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) \cdot (7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}.
\]

Cancel common terms between the numerator and denominator:
- \( 5! \) cancels between \( 6! \) and \( 5! \).
- The \( 6 \) in \( 6! \) cancels with \( 6 \) in \( 7! \).
- Remaining \( 4! \) in the numerator cancels partially with \( 7 \) in \( 7! \).

After simplification:
\[
\frac{6! \cdot 4!}{5! \cdot 7!} = \frac{1}{7}.
\]

### Final Answer:
\[
\boxed{\frac{1}{7}}
\]

Let me know if you'd like additional explanations! 

**Follow-Up Questions:**
1. How are factorials defined?
2. What other operations can factorials simplify?
3. How can combinations and permutations use factorials?
4. What happens if you switch the numerator and denominator?
5. Can factorial expressions be graphed?

**Tip:** Always cancel common terms before expanding factorials fully to simplify computations.

Evaluate \(\frac{6! \cdot 4!}{5! \cdot 7!}\). Simplify your answer as much as possible.

Step-by-step solution to simplify the factorial expression \(\frac{6! \cdot 4!}{5! \cdot 7!}\), covering key concepts such as factorial properties and simplification techniques. Ideal for students in Grades 9-11.

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