The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). It can be defined as:

\[
n! = n \times (n-1) \times (n-2) \times \dots \times 1
\]

For example:
- \( 0! = 1 \) (by definition)
- \( 1! = 1 \)
- \( 2! = 2 \times 1 = 2 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Factorial is widely used in permutations, combinations, and other areas of mathematics involving discrete structures.

Do you want any further details or have specific questions?

Here are 5 questions to expand on this topic:
1. What is the factorial of 5?
2. How is factorial used in combinatorics?
3. Can you explain the relationship between factorials and permutations?
4. How do you calculate \( n! \) for very large values of \( n \)?
5. How is the factorial function related to the gamma function?

**Tip:** The factorial grows extremely fast, making it useful for counting the number of ways to arrange or select items.

Explore the concept of factorial (n!) with a clear definition, key formulas, and examples, such as 3! = 6 and 4! = 24. Learn how factorials are used in combinatorics, permutations, and advanced mathematical fields. Suitable for Grades 8-12.

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