Calculating the factorial of a number is a fundamental concept in mathematics and computer science. The factorial of a non-negative integer $n$, denoted as $n!$, is the product of all positive integers less than or equal to $n$. This article will delve into the definition, properties, and various methods for calculating factorials, including iterative and recursive approaches, as well as practical applications.

## What is a Factorial?

The factorial of a non-negative integer $n$ is defined as: $n!=n×(n−1)×(n−2)×…×1$ For $n=0$, the factorial is defined as $1$: $0!=1$

### Examples

- $5!=5×4×3×2×1=120$
- $3!=3×2×1=6$
- $1!=1$

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## Properties of Factorials

**Factorial of Zero**: $0!=1$ by definition.**Recursive Property**: $n!=n×(n−1)!$.**Growth Rate**: Factorials grow extremely fast. For example, $10!=3,628,800$, and $20!$ exceeds $2.4×1_{18}$.

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## Factorial Calculator

## Practical Applications of Factorials

**Combinatorics**: Factorials are used in permutations and combinations to calculate possible arrangements and selections.**Probability**: Factorials are used in calculating probabilities, particularly in binomial distributions and Bayesian inference.**Algorithms**: Factorials are employed in various algorithms, especially in dynamic programming and backtracking problems.**Mathematical Analysis**: Factorials appear in series expansions, such as the Taylor and Maclaurin series.

## Conclusion

Understanding how to calculate factorials is essential for many areas of mathematics and computer science. Whether using an iterative or recursive approach, or leveraging built-in functions and optimization techniques like memoization, there are multiple ways to compute factorials effectively. With their wide range of applications, mastering the calculation of factorials is a valuable skill in various technical fields.