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.