**Instructions:**

- Enter a non-negative integer in the input field.
- Click "Calculate Factorial" to calculate the factorial.
- View the result, detailed calculation, and formula.
- Your calculation history will be displayed below.
- Click "Clear Results" to reset the results and history.
- Click "Copy Results" to copy the result and explanation to the clipboard.

**Calculation History:**

Factorials, denoted by n!, represent the product of all positive integers from 1 to n. This mathematical concept finds applications in various fields, including combinatorics, probability, and statistics. Understanding factorials and their properties is essential for solving a wide range of mathematical problems.

### The Essence of Factorials: Terminology and Formula

**Factorial:** The factorial of a non-negative integer n is the product of all positive integers from 1 to n. It is denoted by n!, where n is a non-negative integer.

**Formula:** The factorial of a non-negative integer n can be calculated using the following formula:

```
n! = 1 * 2 * 3 * ... * n
```

where n is a non-negative integer.

**Special Cases:**

- 0! = 1 (by convention)
- 1! = 1

### Navigating the Laws of Factorials: Properties and Simplification Rules

Factorials adhere to specific rules that govern their manipulation and simplification. These properties are essential for solving mathematical problems involving factorials.

**Product of Factorials with the Same Base:**

```
a^m * a^n = a^(m + n)
```

**Power of a Factorial:**

```
(a^m)^n = a^(m * n)
```

**Factorial of a Product:**

```
(a * b)^n = a^n * b^n
```

**Quotient of Factorials with the Same Base:**

```
a^m / a^n = a^(m - n)
```

### Benefits of Factorials: Applications and Advantages

Factorials offer numerous benefits and advantages in various mathematical and scientific fields:

**Combinatorics:**Factorials are crucial in combinatorics, the study of arrangements and combinations of objects. They are used to calculate the number of ways to arrange or select objects from a larger set.**Probability:**Factorials are fundamental in probability theory, particularly in discrete probability distributions. They are used to calculate the probability of specific events occurring.**Statistics:**Factorials are employed in statistical analysis, particularly in hypothesis testing and statistical inference. They are used to calculate p-values and confidence intervals.

### Intriguing Facts and Applications of Factorials

- Factorials grow rapidly with increasing n values. For instance, 10! is approximately 3.6288 x 10^6.
- Factorials are used to approximate the number of permutations and combinations of large sets of objects.
- Factorials are used in algorithms for generating random numbers and shuffling data structures.

### References

- “Concrete Mathematics” by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik (1994)
- “Introduction to Probability” by Joseph K. Blitzstein and Jessica Hwang (2014)
- “Combinatorics and Probability” by Graham R. Brightwell and Timothy J. Ott (2009)