10! = 3628800,\quad 3! = 6,\quad 5! = 120,\quad 2! = 2 - Aurero

Understanding the Context

What is a Factorial?

A factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers from 1 to \( n \). Mathematically:

\[
n! = n \ imes (n-1) \ imes (n-2) \ imes \cdots \ imes 2 \ imes 1
\]

For example:
- \( 1! = 1 \)
- \( 2! = 2 \ imes 1 = 2 \)
- \( 3! = 3 \ imes 2 \ imes 1 = 6 \)
- \( 5! = 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 120 \)
- \( 10! = 10 \ imes 9 \ imes 8 \ imes 7 \ imes 6 \ imes 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 3,628,800 \)

Key Insights

Factorials grow very quickly, which makes them powerful in applications involving counting and arrangements.

Factorial Values You Should Know

Here’s a quick look at the key factorials covered in this article:

| Number | Factorial Value |
|--------|-----------------|
| 1! | 1 |
| 2! | 2 |
| 3! | 6 |
| 5! | 120 |
| 10! | 3,628,800 |

Final Thoughts

Step-by-Step Breakdown of Key Factorials

2! = 2
\[
2! = 2 \ imes 1 = 2
\]
The smallest non-trivial factorial, often used in permutations and combinations.

3! = 6
\[
3! = 3 \ imes 2 \ imes 1 = 6
\]
Used when arranging 3 distinct items in order (e.g., 3 books on a shelf).

5! = 120
\[
5! = 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 120
\]
Common in problems involving ways to arrange 5 items, or factorials appear in Stirling numbers and Taylor series expansions.

10! = 3,628,800
\[
10! = 10 \ imes 9 \ imes 8 \ imes 7 \ imes 6 \ imes 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 3,628,800
\]
A commonly referenced large factorial, illustrating the rapid growth of factorial values.