Factorial
Factorial of a whole number 'n' is defined as the product of that number with every whole number till 1. For example, the factorial of 4 is 4×3×2×1, which is equal to 24. It is represented using the symbol '!' So, 24 is the value of 4! In the year 1677, Fabian Stedman, a British author, defined factorial as an equivalent of change ringing. Change ringing was a part of the musical performance where the musicians would ring multiple tuned bells. And it was in the year 1808, when a mathematician from France, Christian Kramp, came up with the symbol for factorial: n! The study of factorials is at the root of several topics in mathematics, such as the number theory, algebra, geometry, probability, statistics, graph theory, and discrete mathematics, etc.
Thinking about how to calculate the factorial of a number? Let's learn.
What Is Factorial?
The factorial of a number is the function that multiplies the number by every natural number below it. Symbolically, factorial can be represented as "!". So, n factorial is the product of the first n natural numbers and is represented as n!
So n! or "n factorial" means: n! = 1. 2. 3…………………………………n = Product of the first n positive integers = n(n1)(n2)…………………….(3)(2)(1)
For example, 4 factorial, that is, 4! can be written as: 4! = \(4 \times 3 \times 2 \times 1\) = 24.
Observe the numbers and their factorial values given in the following table. To find the factorial of a number, multiply the number with the factorial value of the previous number. For example, to know the value of 6! multiply 120 (the factorial of 5) by 6, and get 720. For 7! multiply 720 (the factorial value of 6) by 7, to get 5040.
n  n!  
1  1  1  1 
2  2 × 1  = 2 × 1!  = 2 
3  3 × 2 × 1  = 3 × 2!  = 6 
4  4 × 3 × 2 × 1  = 4 × 3!  = 24 
5  5 × 4 × 3 × 2 × 1  = 5 × 4!  = 120 
Formula for n Factorial
The formula for n factorial is: \[n! = n \times (n  1)!\]
\[n! = n \times (n  1)!\]
This means that the factorial of any number is, the given number, multiplied by the factorial of the previous number. So, \(8! = 8 \times 7!\)...... And \(9! = 9 \times 8!\)...... The factorial of 10 will be \(10! = 10 \times 9!\)...... Like this if we have (n+1) factorial then it can be written as, \((n+1)! = (n+1) \times n!\)
What Is 0!
Zero factorial or Factorial of 0 is interesting, and its value is equal to 1, i.e., 0! = 1
Let us see that how this works:
\(\begin{array}{l}
1! = 1 \\
2! = 2 \times 1 = 2 \\
3! = 3 \times 2 \times 1 = 3 \times 2! = 6 \\
4! = 4 \times 3 \times 2 \times 1 = 4 \times 3! = 24 \\
5! = 5 \times 4 \times 3 \times 2 \times 1 = 5 \times 4! = 120 \\
\end{array}\)
Let’s go to the basic formula of factorial \(n! = n \times (n  1)!\) How to find 4! What you do is \(\dfrac{{5!}}{5}\). Now, let’s look at the pattern:
Factorial of Negative Numbers
Can we have factorials for numbers like −1, −2, etc? Let's start with 3! = \(3 \times 2 \times 1 = 6\)
\(\begin{array}{l}
{\rm{\text{Let's start with } \ 3!= 3 \times 2 \times 1 \\ = 6\: \text{and go down }:}} \\
{\rm{2! = 3! / 3 = 6 / 3 = 2}}\\
{\rm{1! = 2! / 2 = 2 / 2 = 1}} \\
{\rm{0! = 1! / 1 = 1 / 1 = 1 }}\\
\left( {{\rm{  1}}} \right){\rm{! = 0! / 0 = 1 / 0 \\= {\text{ dividing by zero is undefined}}}} \\
\end{array}\)
And from here on down all integer factorials are undefined. So, negative integer factorials are undefined.
Use of Factorial
One area where factorials are commonly used is in permutations & combinations. Now, permutation is an ordered arrangement of outcomes and it can be calculated with the formula:
^{n}P_{r} = \(\dfrac{{n!}}{{(n  r)!}}\)
Combination is a grouping of outcomes in which the order does not matter. It can be calculated with the formula:
^{n}C_{r} = \(\dfrac{{n!}}{{r!(n  r)!}}\)
Let us understand this by the following examples.
Example 1
In a group of 10 people, $200, $100, and $50 prizes are to be given. In how many ways can the prizes be distributed?
Solution:
This is permutation because here the order matters. It can be calculated as \(^{{\rm{1}}0}{{\rm{P}}_{\rm{3}}}\) ways.
\(^{{\rm{1}}0}{{\rm{P}}_{\rm{3}}} = \dfrac{{10!}}{{(10  3)!}} = \dfrac{{10!}}{{7!}} \\= \dfrac{{10 \times 9 \times 8 \times 7!}}{{7!}} = 720\) ways.
Example 2
Three $50 prizes are to be distributed in a group of 10 people. In how many ways can the prizes be distributed?
Solution:
This is a combination because here the order does not matter. It can be calculated as, \(^{{\rm{1}}0}{{\rm{C}}_{\rm{3}}}\) ways.
\(^{{\rm{1}}0}{{\rm{C}}_{\rm{3}}} = \dfrac{{10!}}{{3!(10  3)!}} = \dfrac{{10!}}{{3!7!}} \\= \dfrac{{10 \times 9 \times 8 \times 7!}}{{3 \times 2 \times 1 \times 7!}} = 120\) ways.
Calculation of Factorial
The factorial of n is denoted by n! and calculated by the integer numbers from 1 to n. The formula for n factorial is \(n! = n \times (n  1)!\).
Example
If 8! is 40,320 then what is 9!?
Solution
9! = 9 × 8! = 9 × 40,320 = 362,880
Now, let us look at a factorial table given below that shows the values of factorial for the first 15 natural numbers:
n  n! 
1  1 
2  2 
3  6 
4  24 
5  120 
6  720 
7  5040 
8  40,320 
9  362,880 
10  3,628,800 
11  39,916,800 
12  479,001,600 
13  6,227,020,800 
14  8,717,8291,200 
15  1,307,674,368,000 
Topics Related to Factorial
Click on these articles to know more about factorial in math!
 Factorial Calculator
 What is the Factorial of 9?
 What is the Factorial of 20?
 Permutations
 Combinations
Important Notes:
 The factorial of any whole number can be calculated as \(n! = n \times (n  1)!\).
 The value of zero factorial is one, i.e., 0! = 1. Negative integer factorials are undefined.
 Permutation & Combination can be calculated as, ^{n}P_{r} = \(\dfrac{{n!}}{{(n  r)!}}\) &^{ n}C_{r} = \(\dfrac{{n!}}{{r!(n  r)!}}\) respectively.
Solved Examples on Factorial

Example 1: Evaluate the expression \(\dfrac{{10!}}{{4! \times 6!}}\)
Solution:\(\begin{array}{l}
\dfrac{{10!}}{{4! \times 6!}} &= \dfrac{{10 \times 9 \times 8 \times 7 \times 6!}}{{4! \times 6!}} \\[0.2cm]
&= \dfrac{{10 \times 9 \times 8 \times 7}}{{4 \times 3 \times 2 \times 1}} \\[0.2cm]
&= 210 \\
\end{array}\)Therefore, the value of expression \(\dfrac{{10!}}{{4! \times 6!}}\) is 210.

Example 2: Find the value of 5! (6  3)!
Solution:5! (6−3)! = 5!× 3!
=(5×4×3×2×1)(3×2×1)
=120×6
=720
Therefore, the value of 5! (6  3)! is 720. 
Example 3: How many 5digit numbers can be formed using the digits 1, 2, 5, 7, and 8 in each of which no digit is repeated?
Solution:The given 5 digits (1, 2, 5, 7 and 8) should be arranged among themselves in order to get 5digit numbers.
The number of ways for doing this can be done using the factorial.
5! = 5 × 4 × 3 × 2 × 1 = 120
Therefore, the required number of 5digit numbers is 120. 
Example 4: In how many ways can eight people line up from left to right for a group photo?
Solution: Number of ways 8 people line up
= 8!
= 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 40,320
Therefore, Eight people can line up in 40,320 ways.
Practice Questions on Factorial
FAQs on Factorial
What is the Relation Between Factorial, Permutation, and Combination?
Factorial is a function which 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.
What is n+1 Factorial?
n+1 factorial can be calculated as (n+1)! = (n+1)n!
What is the Factorial of 10?
10! can be calculated as 10!=10×9!=10×362,880=3,628,800.
What is the Factorial of a Number?
In mathematics, factorial of a number means multiplication of a positive integer with every integer less than that. So, n!= n × (n1) × (n2) × (n3) × ..... × 3 × 2 × 1.
What are Factorials Used for?
Factorials are used to find the number of patterns, solving permutation and combination problems, finding out probability of events, etc.
What is the Factorial Symbol?
The symbol used to represent factorial is ' ! '.
What is Factorial Notation?
Factorial notation is the expanded form of the factorial of a number. So, 3! is 3 × 2 × 1, 5! is 5×4×3×2×1, and so on.
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