# Factorial Calculator

#### Factorial

Have you ever thought of how many ways you can arrange a set of books on your shelf? Or perhaps, how different team lineups can be formed from a group of players? The answer lies in the mathematical concept known as the factorial. This intriguing operation, denoted by an exclamation mark (!), is not just a mere symbol but a gateway to permutations, combinations, and beyond. Factorials play a crucial role in various fields from computer algorithms to statistical calculations and make them an essential tool for anyone probing into the mathematical or scientific problems.

## Table of Contents

First of all, you will be eager to know

**How to Use a Factorial Calculator**

For your convenience, say, Calculate 6!

Steps:

- Input ‘n’: = 6
- Click CALCULATE.

Observe the result: 6! = 720

Tips Tricks to Use Factorial Calculator

In order that you could not face any problem while using the factorial calculator, follow the tips and tricks:

- Don’t use negative number.
- Don’t use mixed number.
- Don’t use decimals.
- Don’t use “!” in the box.

Well, probably you are in a fix how it becomes 720? To meet your curiosity, here is the detail.

**What is a Factorial?**

Factorial might sound complicated, but it’s actually a simple concept. The factorial of a non-negative number, written as *n*! is just the result of multiplying all positive whole numbers up to *n*. For instance, the factorial of 5 (5!) is calculated as 5 x 4 x 3 x 2 x 1, which equals 120. Mathematically, it is expressed as n!=n×n-1n-2×… ×3×2×1.

*Factorial Formula:*

*Factorial Formula:*

A. Standard Formula: n!=n×n-1!

B. Base Case: 0!=1.

- 1! = 1.
- 2! = 2.

**Sub Factorial of a Number**

The sub factorial, also known as the derangement number or !*n* (pronounced “sub factorial *n*“), represents the number of permutations of a set where none of the elements appear in their original position. It is denoted as !*n*.

For example, if you have the set {1, 2, 3}, the permutations include {2, 3, 1}, {3, 1, 2}, and so on. A derangement occurs when none of the elements are in their original positions.

The formula for the sub factorial is:

\[ !n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!} \]

For example:

- For
*n*= 3; 3! = 6

Sub factorial: !3 = 2 because there are two derangements:

{2, 3, 1} and {3, 1, 2}.

- For
*n*= 4; 4! = 24

Sub factorial: !4 = 9 because there are nine derangements:

{2, 1, 4, 3},

{2, 3, 4, 1},

{2, 4, 1, 3},

{3, 4, 1, 2},

{4, 1, 2, 3},

{4, 3, 1, 2},

{1, 4, 2, 3},

{3, 1, 4, 2}, and

{1, 3, 4, 2}.

**Factorial in Calculator n choose k**

To calculate \( \binom{n}{k} \) where ‘*n*‘ is the total number of elements and ‘k’ is the number of elements to be chosen. The formula involves factorials:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

\(\text{Example: Calculate } \binom{5}{2} \)

\(\binom{5}{2} = \frac{5!}{2!(5-2)!}\)

\(= \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times (3 \times 2 \times 1)}\)

\(= \frac{5 \times 4}{2 \times 1}\)

\(= 10\)

**Factorial Calculator ***n* and *r*

*n*and

*r*

- For n! Calculating 4! \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
- For
*n*! and*r*! calculating \( \frac{8!}{3!} \) \[ \frac{8!}{3!} = \frac{40320}{6} = 6720 \]

**Use of Factorial in Real-Life**

- Permutations and Combinations: Factorials model the number of ways objects can be arranged or chosen. It’s applicable in scenarios like arranging books or forming teams.

- Probability: In probability theory, factorials help calculate the likelihood of events, especially when considering different outcomes.

- Statistics: Factorials are used in statistical calculations which contribute to the understanding of distributions and probabilities.

**Why Use Factorial Calculator**

The importance of using a factorial calculator extends beyond mere convenience. Here are some points highlighting why this tool is crucial:

- Time Efficiency: Factorial calculations involving large numbers can be time-consuming when done manually. A factorial calculator provides instant results and saves valuable time, and enables users to focus on interpreting outcomes rather than spending time on tedious computations.

- Reduced Human Error: Manually calculating factorials, especially for extensive sequences, introduces the risk of human error. A factorial calculator eliminates this concern and ensures accuracy in complex calculations and prevents mistakes that might arise from fatigue or oversight.

- Ease of Handling Large Numbers: A factorial calculator effortlessly handles computations for substantial values of
*n*and offers a practical solution for researchers, mathematicians, and professionals dealing with intricate mathematical problems.

- Versatility in Applications: Factorial calculations are integral to various disciplines including mathematics, statistics, and computer science. A factorial calculator’s versatility makes it suitable for a range of applications from solving combinatorial problems to determining probabilities in statistical analyses.

- Enhanced Problem-Solving: By streamlining the calculation process, a factorial calculator empowers users to focus on the application of factorial concepts rather than getting bogged down by intricate arithmetic. This enhanced problem-solving capability is particularly valuable in scenarios where quick and accurate results are essential.

Wrapping up, the factorial calculator serves as a handy tool in simplifying complex mathematical computations involving factorials. Whether calculating permutations, combinations, sub factorials, or probabilities, this calculator enhances efficiency, reduces the risk of human error, and handles large numbers effortlessly. Its versatility extends to applications in various fields such as mathematics, statistics, and computer science. In real-life scenarios, the factorial calculator proves essential which allows users to focus on interpreting outcomes and apply factorial concepts rather than laborious arithmetic.

### FAQ

### What is the value of 0! ?

The value of 0! is 1.

### How do factorials grow as the input number increases?

Factorials grow at an exponential rate.

### How do you find the factorial of a number?

To calculate the factorial of a number n, you just multiply all the positive whole numbers from 1 to n.

### How is factorial denoted in mathematical notation?

Factorial is symbolized by an exclamation mark (!).

### What is the value of 2! ?

The value of 2! is 2.