The expression you provided is:

$\frac{5!}{3!(5 - 3)!}$

Let's solve it step by step without using a calculator.

Step 1: Simplify the terms

First, let's evaluate the factorials:

• $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
• $3! = 3 \times 2 \times 1 = 6$
• $5 - 3 = 2$, so $2! = 2 \times 1 = 2$

Now substitute these into the expression:

$\frac{5!}{3!(5 - 3)!} = \frac{120}{6 \times 2}$

Step 2: Perform the multiplication in the denominator

$6 \times 2 = 12$

Step 3: Simplify the fraction

$\frac{120}{12} = 10$

So, the answer is:

$\boxed{10}$

Would you like more details on any step? Here are some related questions:

1. What is the general formula for combinations involving factorials?
2. Can you explain what factorials represent in probability?
3. How do you simplify other expressions involving multiple factorials?
4. How do binomial coefficients relate to factorials?
5. What is the significance of $n!$ in different areas of math?

Tip: Factorials grow very quickly, so it's helpful to know how to break them down into smaller products to simplify expressions.