What is Factorial of a Number – Definition, Formula & Examples

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 $0 = 0! = 1$

Read Why Is 0 Factorial 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

1. Evaluate the following
   • $4!$
   • $5! \times 4!$
   • $7! \times 0!$
   • $\frac{4!}{0!}$
   • $\frac{6!}{2! \times 4!}$
2. Simplify the following expressions
   • $\frac{(n + 2)!}{n!}$
   • $\frac{(2n + 2)!}{2n!}$
   • $\frac{(n – 1)!}{(n + 1)!}$
   • $\frac{(n + 1)!}{ n!}$

FAQs

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.

Recommended Reading

• Fundamental Principle of Counting – Rules, Formulas & Examples
• Why Is 0 Factorial 1?