https://file.aiquickdraw.com/mathbot/file/1728456347995eyamhf2j.jpg

The image shows two expressions that require the evaluation of factorials. Let's solve them step by step.

### (a) \( \frac{8!}{5!} \)

First, we recall the definition of factorial:

- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

Notice that the \( 5! \) terms will cancel out, leaving:

\[
\frac{8!}{5!} = \frac{8 \times 7 \times 6 \times (5!)}{5!} = 8 \times 7 \times 6 = 336
\]

### (b) \( \frac{8!}{3!} \)

Similarly:

- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)
- \( 3! = 3 \times 2 \times 1 \)

Cancel out the \( 3! \) terms:

\[
\frac{8!}{3!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times (3!)}{3!} = 8 \times 7 \times 6 \times 5 \times 4 = 6720
\]

So, the answers are:
- (a) \( \frac{8!}{5!} = 336 \)
- (b) \( \frac{8!}{3!} = 6720 \)

Would you like any further details or explanations?

### Related questions:
1. How is the factorial operation defined for \( n! \)?
2. What is the significance of factorials in combinatorics and probability theory?
3. Can you derive a general formula for \( \frac{n!}{(n-k)!} \)?
4. How do factorial operations apply in binomial expansions?
5. How can you simplify factorial expressions when given complex ratios like \( \frac{n!}{(k!)^2} \)?

### Tip:
Factorials grow extremely fast. For example, \( 10! = 3,628,800 \), so it's helpful to simplify terms whenever possible.

Find the indicated values of the expressions (a) 8!/5! and (b) 8!/3!.

Learn how to simplify the factorial expressions 8!/5! and 8!/3! with a step-by-step explanation. This problem covers key mathematical concepts involving factorials and simplification techniques, suitable for students in Grades 9-10.

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