To solve the expression $\frac{10!}{(10-3)! \cdot 3!}$, we can simplify it step-by-step.

Step 1: Understand the Expression

The expression given is:

$\frac{10!}{(10-3)! \cdot 3!}$

First, simplify the expression inside the parentheses:

$10 - 3 = 7$

So, the expression becomes:

$\frac{10!}{7! \cdot 3!}$

Step 2: Simplify the Factorial Expression

Recall that $n! = n \times (n-1) \times (n-2) \times \ldots \times 1$.

We can expand $10!$ as:

$10! = 10 \times 9 \times 8 \times 7!$

Thus, the expression can be rewritten as:

$\frac{10 \times 9 \times 8 \times 7!}{7! \cdot 3!}$

The $7!$ in the numerator and the denominator cancel out, leaving:

$\frac{10 \times 9 \times 8}{3!}$

Step 3: Compute $3!$

Calculate $3!$:

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

Step 4: Divide and Simplify

Now, substitute $3!$ back into the expression:

$\frac{10 \times 9 \times 8}{6}$

Let's compute the numerator:

$10 \times 9 = 90$ $90 \times 8 = 720$

So, the expression is:

$\frac{720}{6}$

Now, divide:

$\frac{720}{6} = 120$

Final Answer

The value of the expression $\frac{10!}{(10-3)! \cdot 3!}$ is:

$\boxed{120}$

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. What is the value of $\frac{8!}{(8-2)! \cdot 2!}$?
2. How do you calculate combinations $\binom{n}{r}$ using factorials?
3. What is the difference between permutations and combinations?
4. Can factorial expressions be simplified further in different cases?
5. How does the factorial function grow as $n$ increases?

Tip: When simplifying factorials in expressions, look for terms that cancel out to make calculations easier.