https://file.aiquickdraw.com/mathbot/file/1726841092593mknzbfq5.jpg

The expression in the image is:

\[
\frac{8!4!}{7!2!}
\]

Let’s break it down step-by-step to simplify it:

1. **Factorials**:
   - \( 8! = 8 \times 7! \)
   - \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
   - \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \), but we'll leave it as \( 7! \) for now to simplify later.
   - \( 2! = 2 \times 1 = 2 \)

2. **Substitute these into the expression**:
   \[
   \frac{8!4!}{7!2!} = \frac{8 \times 7! \times 24}{7! \times 2}
   \]

3. **Cancel out \( 7! \) from both the numerator and denominator**:
   \[
   = \frac{8 \times 24}{2}
   \]

4. **Simplify**:
   \[
   = \frac{192}{2} = 96
   \]

So, the simplified answer is:

\[
\boxed{96}
\]

Would you like further details or have any questions about this process?

Here are 5 related questions to expand on this topic:
1. What is the factorial of any number \( n \) and how is it calculated?
2. How do factorials relate to combinatorics or permutations?
3. Can factorials grow exponentially, and what implications does that have?
4. What happens when you divide large factorials, and how can they be simplified efficiently?
5. Can factorials be applied in probability problems like combinations and permutations?

**Tip**: When simplifying factorials in fractions, cancel out common terms between the numerator and the denominator first to make the calculation easier.

Learn how to simplify the expression 8!4! / 7!2! with a detailed step-by-step explanation. This problem involves key concepts like factorial simplification and algebraic fraction reduction. Suitable for high school students in Grades 9-12.

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