In mathematics, "(!)" represents the factorial operation. The factorial of a non-negative integer is the product of all positive integers less than or equal to that number.
Understanding Factorials
Factorials are denoted by the exclamation mark (!). According to the reference, a factorial means to multiply by decreasing positive integers.
How to Calculate Factorials
To calculate the factorial of a number, you multiply that number by every positive integer smaller than it, down to 1.
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For example, 5! (read as "5 factorial") is calculated as:
5! = 5 4 3 2 1 = 120
Examples of Factorials
Here are a few more examples to illustrate the factorial operation:
- 3! = 3 2 1 = 6
- 4! = 4 3 2 * 1 = 24
- 6! = 6 5 4 3 2 * 1 = 720
Important Note: 0!
By definition, 0! (zero factorial) is equal to 1. This might seem counterintuitive, but it's a necessary convention for many mathematical formulas to work correctly, especially in combinatorics.
Factorials in Combinatorics and Probability
Factorials are essential in combinatorics and probability for calculating permutations (the number of ways to arrange things in a specific order) and combinations (the number of ways to choose things without regard to order). For example, the number of ways to arrange n distinct objects is n!.
Summary Table
| Expression | Calculation | Result |
|---|---|---|
| 3! | 3 2 1 | 6 |
| 4! | 4 3 2 * 1 | 24 |
| 5! | 5 4 3 2 1 | 120 |
| 6! | 6 5 4 3 2 * 1 | 720 |
| 0! | Defined as | 1 |
