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.

Learn how to simplify the factorial expression 10! / ((10-3)! * 3!) step-by-step with a detailed solution. This problem involves factorials, combinations, and division. Suitable for students in Grades 10-12.

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