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.