https://file.aiquickdraw.com/mathbot/file/172978511440150mlh79h.jpg

The expression shown in the image is:

\[
\frac{9!}{4! \cdot 5!}
\]

This expression can be solved using factorials. Factorial notation (e.g., \( n! \)) means multiplying all whole numbers from \( n \) down to 1. Let's break it down step by step:

### Step 1: Calculate the factorials
- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362,880 \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

### Step 2: Multiply the factorials in the denominator
\[
4! \cdot 5! = 24 \cdot 120 = 2,880
\]

### Step 3: Divide the factorials
\[
\frac{9!}{4! \cdot 5!} = \frac{362,880}{2,880} = 126
\]

### Final Answer:
\[
\frac{9!}{4! \cdot 5!} = 126
\]

Would you like a further breakdown or explanation on any of the steps?

Here are some related questions:
1. What is the general formula for combinations \( \binom{n}{k} \), and how does it relate to this problem?
2. How does the factorial function grow as numbers increase?
3. Can you calculate \( \frac{10!}{5! \cdot 5!} \) using a similar method?
4. How would you simplify a similar expression like \( \frac{8!}{3! \cdot 5!} \)?
5. Why do we use factorials in combinatorics and probability?

**Tip**: Factorials grow extremely fast as the number increases, making them useful in counting large possibilities, such as in permutations and combinations.

What is the value of the expression \( \frac{9!}{4! \cdot 5!} \)?

Learn how to solve the factorial expression \( \frac{9!}{4! \cdot 5!} \) with a step-by-step solution. This problem involves calculating large factorials and simplifying them, which is essential in combinatorics and probability. Suitable for students in Grades 9-12.

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