The image contains a mathematical problem asking to compute several factorial expressions. Let me solve each part for you.

Problem Breakdown

The problem asks to compute the values for the following expressions:

a. $3!$
b. $4! - 2!$
c. $6! \times 2$
d. $\frac{8!}{5! \times (8-5)!}$
e. $\frac{10!}{9!} \times 2!$

Let's calculate each part one by one.

Solutions:

a. $3! = 3 \times 2 \times 1 = 6$

b.
$4! = 4 \times 3 \times 2 \times 1 = 24$
$2! = 2 \times 1 = 2$
So, $4! - 2! = 24 - 2 = 22$

c.
$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
$6! \times 2 = 720 \times 2 = 1440$

d.
The expression can be simplified as:
$\frac{8!}{5! \times (8-5)!} = \frac{8!}{5! \times 3!}$
Now, compute the factorials:
$8! = 8 \times 7 \times 6 \times 5!$
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
$3! = 3 \times 2 \times 1 = 6$

So:
$\frac{8!}{5! \times 3!} = \frac{8 \times 7 \times 6 \times 5!}{5! \times 6} = \frac{8 \times 7}{1} = 56$

e.
$\frac{10!}{9!} \times 2! = (10 \times 9!) / 9! \times 2! = 10 \times 2!$
Since $2! = 2$,
$10 \times 2 = 20$

Final Answers:

• a. 6
• b. 22
• c. 1440
• d. 56
• e. 20

Would you like more details on any of these calculations or have any questions?

Related Questions:

1. What are factorials, and why are they used in mathematics?
2. How do combinations and permutations differ, and where do factorials come into play?
3. What is the relationship between factorials and exponential functions?
4. How can factorial calculations be optimized for larger numbers?
5. What are some real-world applications of factorials in probability and statistics?

Tip: Understanding the properties of factorials can help simplify complex combinatorial problems.