In Mathematics, factorial is an important function, which is used to find how many ways things can be arranged or the ordered set of numbers. The factorial of a whole number $n$ is defined as the product of that number with every whole number less than or equal to $n$ till $1$. Starting in the 1200s, factorials were used to count permutations. The notation for a factorial ($n!$) was introduced in the early 1800s by Christian Kramp, a French mathematician.

Let’s understand what is factorial of a number and how it is calculated with examples.

## What is Factorial of a Number?

In mathematics, the factorial of a whole number is the function that multiplies the number by every natural number less than it. A factorial is represented by the symbol “$!$”. $n$ factorial is represented as $n!$ and is the product of the first $n$ natural numbers. So, $n!$ is equal to $1 \times 2 \times 3 \times 4 … \times (n – 1) \times n$.

**Note:**

- Factorial of $1 = 1! = 1$
- Factorial of $0 = 0! = 1$

### Examples of Factorial of a Number

**Example 1:** Find $5!$.

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

**Example 2:** Find $8!$.

$8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320$.

## Factorial of a Number Formula

To understand the formula of a factorial of a number, let’s create a table as shown below.

$n$ | $n!$ | Result |

$1$ | $1! $ | $1$ |

$2$ | $2! = 2 \times 1$ | $2 = 2 \times 1!$ |

$3$ | $3! = 3 \times 2 \times 1$ | $6 = 3 \times 2!$ |

$4$ | $4! = 4 \times 3 \times 2 \times 1$ | $24 = 4 \times 3!$ |

$5$ | $5! = 5 \times 4 \times 3 \times 2 \times 1$ | $120 = 5 \times 4!$ |

$6$ | $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$ | $720 = 6 \times 5!$ |

Now, observe the numbers and their factorial values given in the above table. You will notice that the factorial of a number is obtained by multiplying the number with the factorial value of the previous number(Result column).

For example, to get the value of $5!$, $5$ is multiplied by $24$(i.e., $4!$) and to get $6!$, $6$ is multiplied by $120$(i.e., $5!$). It means to get the factorial of a number, the number is multiplied by the factorial of the previous number.

Mathematically, it can be represented as $n! = n \times (n – 1)!$.

## Factorials of First $10$ Natural Numbers

The factorial values for the first natural numbers are

$n$ | $n!$ |

$1$ | $1! = 1$ |

$2$ | $2! = 2$ |

$3$ | $3! = 6$ |

$4$ | $4! = 24$ |

$5$ | $5! = 120$ |

$6$ | $6! = 720$ |

$7$ | $7! = 5040$ |

$8$ | $8! = 40320$ |

$9$ | $9! = 362880$ |

$10$ | $!10 = 3628800$ |

**Examples of** **Factorial of a Number Formula**

**Example 1:** What is the factorial of $10$?

We know that the factorial formula is $n! = n \times (n – 1) \times (n – 2) \times (n – 3) \times …. \times 3 \times 2 \times 1$.

So the factorial of $10$ is $10! = 10 \times (10 -1) \times (10 – 2) \times (10 – 3) \times (10 – 4) \times (10 – 5) \times (10 – 6) \times (10 – 7) \times (10 – 8) \times 1$

$10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800$

Therefore, the factorial of $10$ is $3628800$.

**Example 2: **Solve for $\frac{9! \times 8!}{ 7! \times 6!}$

$\frac{9! \times 8!}{ 7! \times 6!}$

$=\frac{(9 \times 8 \times 7! ) \times (8 \times 7 \times 6! )}{ 7! \times 6!}$

Canceling $7!$ and $6!$, we get $\frac{(9 \times 8) \times (8 \times 7 )}{ 1}$

$9 \times 8 \times 8 \times 7 = 4032$.

**Example 3:** Solve $\frac{n!}{r!(n-r)!}$ When $n = 9$ and $r = 8$

Substituting $n = 9$ and $r = 8$ in $\frac{n!}{r!(n-r)!}$, we get

$\frac{9!}{8!(9 – 8)!} = \frac{9!}{8! \times 1!} = \frac{9!}{8! \times 1}$

$\frac{9!}{8!} = \frac{9 \times 8!}{8!} = 9$.

**Example 4:** If $n! = 24$, find $n$.

We know that by the formula of factorial $n! = 1 \times 2 \times 3 \times ….(n-1) \times n!$

Given that $n! = 24$

Therefore, $n \times (n – 1) \times (n – 2) \times 1 = 4 \times 3 \times 2 \times 1$

$=> n = 4$.

**Example 5:** Is $10! = 8! + 2!$?

LHS = $10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800$.

RHS = $8! + 2! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 2 \times 1 = 40320 + 2 = 40322$.

Therefore, $10! \ne 8! + 2!$.

**Example 6: **If $(n+1)! = 6n!$, find $n$.

$(n+1)! = 6n! => (n+1) \times n! = 6n! => (n+1) \times n! = 6 \times n! => n+1 = 6 => n = 5$.

## What is Sub-Factorial of a Number?

A mathematical term sub-factorial, defined by the term $!n$, is defined as the number of rearrangements of $n$ objects. It means that the number of permutations of $n$ objects so that no object stands in its original position. The formula to calculate the sub-factorial of a number is given by $!n = n! \sum{\frac{(-1)^k}{k!}}$,

The sub-factorial values for the first natural numbers are

$n$ | $!n$ |

$1$ | $!1 = 0$ |

$2$ | $!2 = 1$ |

$3$ | $!3 = 2$ |

$4$ | $!4 = 9$ |

$5$ | $!5 = 44$ |

$6$ | $!6 = 265$ |

$7$ | $!7 = 1854$ |

$8$ | $!8 = 14833$ |

$9$ | $!9 = 133496$ |

$10$ | $!10 = 1334961$ |

## Use of Factorial of a Number

In mathematics, factorials are commonly used in permutations and combinations.

- Permutation is an ordered arrangement of outcomes and it can be calculated with the formula $^{n}\text{P}_{r} = \frac{n!}{(n – r)!}$
- Combination is a grouping of outcomes in which order does not matter. It can be calculated with the formula $^{n} \text{C}_{r} = \frac{n!}{r!(n – r)!}$

In both of these formulas, $n$’ is the total number of things available, and $r$ is the number of things that have been chosen.

## Practice Problems

- Evaluate the following
- $4!$
- $5! \times 4!$
- $7! \times 0!$
- $\frac{4!}{0!}$
- $\frac{6!}{2! \times 4!}$

- Simplify the following expressions
- $\frac{(n + 2)!}{n!}$
- $\frac{(2n + 2)!}{2n!}$
- $\frac{(n – 1)!}{(n + 1)!}$
- $\frac{(n + 1)!}{ n!}$

## FAQs

### What is the factorial of a Number?

The factorial of a whole number is the function that multiplies the number by every natural number less than it. A factorial is represented by the symbol “$!$”. $n$ factorial is represented as $n!$ and is the product of the first $n$ natural numbers. So, $n!$ is equal to $1 \times 2 \times 3 \times 4 … \times (n – 1) \times n$.

### What is the factorial symbol?

The symbol used to represent factorial is ‘$ !$ ‘. For example “5 factorial” is written as 5!.

### What is factorial notation?

Factorial notation is writing the product of consecutive whole numbers in the form of a factorial. It means $n! = n \times (n – 1) \times (n – 2) \times (n – 3) \times (n – 4) \times 1$.

For example, $3! = 3 × 2 × 1$.

### What is the value of $1!$?

The value of $1! = 1$.

### What is the value of $0!$?

The value of $0! = 1$.

Read Why Is 0 Factorial 1?

### Where do we use factorials?

Factorial is a function that is used to find the number of possible ways in which a selected number of objects can be arranged among themselves. This concept of factorial is used for finding permutations and combinations of numbers and events.

## Conclusion

In mathematics, the factorial of a whole number is the function that multiplies the number by every natural number less than it. A factorial is represented by the symbol “$!$”. $n$ factorial is represented as $n!$ and is the product of the first $n$ natural numbers. So, $n!$ is equal to $1 \times 2 \times 3 \times 4 … \times (n – 1) \times n$. The most common applications of factorial are in permutations and combinations.