Factorial Notation: The common use for the factorial function is to count how many ways you can choose things from a collection of things. For example, let’s say you are going on a trip. You own \(p\) shirts but you have a bag to pack only \(n\) of them. How many distinct ways can you choose \(n\) shirts from the collection of \(p\) shirts? The answer is calculated using factorials. In this article, let us learn how to calculate the distinct number of ways to choose shirts for the trip.

The factorials are also used in various other fields of mathematics such as number theory, algebra, geometry, probability, statistics, graph theory, and discrete mathematics.

Formula for Factorial Notation

Factorial of a whole number, \(n\), is defined as the product of that number with every smaller whole number till \(1\).
For example, the factorial of \(3\) is \(3 \times 2 \times 1\) which is equal to \(6\).
The continuous product of first \(n\) natural numbers is called the “\(n\) factorial”. It is denoted by \(n\)!
\(n !=1 \times 2 \times 3 \times 4 \times \ldots \times(n-1) \times n\)
It can also be represented as,

Thus,
\(1 !=1\)
\(2 !=1 \times 2=2\)
\(3 !=1 \times 2 \times 3=6\)
\(4 !=1 \times 2 \times 3 \times 4=24\)
\(5 !=1 \times 2 \times 3 \times 4 \times 5=120\)
\(6 !=1 \times 2 \times 3 \times 4 \times 5 \times 6=720\)
\(7 !=1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7=5040\)
\(8 !=1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8=40320\)
\(9 !=1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9=362880\)
\(10 !=1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10=3628800\)
Note that \(n!\) is defined for positive integers only.

Counting Up and Counting Down

We can calculate the factorial either by counting up or down.
For example, 
\(3 !=1 \times 2 \times 3\)
\(3 !=3 \times 2 \times 1\)

Zero Factorial

Observe that although we can calculate factorial for all whole numbers, we leave out the zero while calculating the factorials of natural numbers. 

\({\rm{1! = 1}}\)
\(2! = 2 \times 1 = 2\)
\(3! = 3 \times 2 \times 1 = 3 \times 2! = 6\)
\(4! = 4 \times 3 \times 2 \times 1 = 4 \times 3! = 24\)
\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 5 \times 4! = 120\)

What do you think is the value of zero factorial \((0!)\)?
Mathematically, the factorial of zero is defined as,
\(0 !=1\)

Let’s see that how this works:

Deduction of the Pattern

We have,
\(n !=1 \times 2 \times 3 \times 4 \ldots \times(n-1) \times n\)
\(=[1 \times 2 \times 3 \times 4 \ldots \times(n-1)] n\)
\(=[(n-1) !] n=n \times(n-1) !\)
Thus, \(n !=n \times(n-1) !\)

The factorial of first \(5\) natural numbers can be written as shown below.

Number	Factorial	Factors	Value
\(1\)	\(1\)	\(1\)	\(1\)
\(2\)	\(2 \times 1\)	\( = 2 \times 1!\)	\(=2\)
\(3\)	\(3 \times 2 \times 1\)	\( = 3 \times 2!\)	\(=6\)
\(4\)	\(4 \times 3 \times 2 \times 1\)	\( = 4 \times 3!\)	\(=24\)
\(5\)	\(5 \times 4 \times 3 \times 2 \times 1\)	\( = 5 \times 4!\)	\(=120\)

Similarly,
\(n !=n(n-1)(n-2) !\)
\(=n(n-1)(n-2)(n-3) !\)
\(=n(n-1)(n-2)(n-3)(n-4) !\)

Factorial of Negative Numbers

Can we have factorials for negative numbers like \(-1,-2,-3\), and \(-4\) ?
Let’s start with \(3!\)
We know that \(3 !=3 \times 2 \times 1\) and \(2 !=2 \times 1\)
\(\therefore 2 !=\frac{3 !}{3}=\frac{6}{3}=2\)
\(\Rightarrow 1 !=\frac{2 !}{2}=\frac{2}{2}=1\)
\(\Rightarrow 0 !=\frac{1 !}{1}=\frac{1}{1}=1\)
This means, we can write \((-1) !\) as,
\((-1) !=\frac{0 !}{0}=\frac{1}{0}\)
Evidently, dividing any number by zero is undefined in Mathematics.
Hence, factorials of negative integers are undefined.

Note: Similarly, factorials of proper fractions are not defined. 

Applications of Factorials

Factorials are commonly used in calculations of permutations and combinations.
A permutation is an ordered arrangement of outcomes and it can be calculated using the formula: 
\(P(n, r)=\frac{n !}{(n-r) !}\)
The combination is a grouping of outcomes in which the order does not matter. It can be calculated using the formula: 
\(C(n, r)=\frac{n !}{r !(n-r) !}\)

Other sequences similar to the factorial are:

• Double Factorials: are used to simplify trigonometric integrals.
• Multi-factorials: can be denoted with multiple exclamation points.
• Primorials: entail getting the product of the prime numbers, which are less than or equal to \(n\).
• Super-factorials: are defined as the product of the first \(n\) factorials.
• Hyper-factorials: are a result of multiplying a number of consecutive values ranging from \(1\) to \(n\).

Solved Examples

Q.1. Find the value of \(6!\)
Ans: We know that, factorial is, 
\(n !=n \times(n-1) \times(n-2) \times(n-3) \times \ldots \times 3 \times 2 \times 1\)
So, \(6 !=6 \times(6-1) \times(6-2) \times(6-3) \times(6-4) \times 1\)
\(=6 \times 5 \times 4 \times 3 \times 2 \times 1\)
\(=720\)
Hence, the required value of \(6!\) is \(720\).

Q.2. Find the value of \(9!\)
Ans: We know that, factorial is,
\(n !=n \times(n-1) \times(n-2) \times(n-3) \times \ldots \times 3 \times 2 \times 1\)
So, \(9 !=9 \times(9-1) \times(9-2) \times(9-3) \times(9-4) \times(9-5) \times(9-6) \times(9-7) \times(9-9) \times 1\)
\(=9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)
\(=362880\)
Hence, the required value of \(9!\) is \(362880\).

Q.3. Find the value of \(n\), if \((n+2) !=2550 \times n !\)
Ans: Given that, we have \((n+2) !=2550 \times n !\)
We know that, factorial is, 
\(n !=n \times(n-1) \times(n-2) \times(n-3) \times \ldots \times 3 \times 2 \times 1\)
\(\therefore(n+2)(n+1) \times n !=2550 \times n !\)
\(\Rightarrow(n+2) \times(n+1)=2550\)
\(\Rightarrow(n+2)(n+1)=51 \times 50\)
On comparing both sides, we have, \(n+2=51\) or \(n+1=50\)
\(\Rightarrow n=49\)
Hence, the required value of \(n\) is \(49\).

Q.4. Find the value of \(n\), if \((n+1) !=12 \times(n-1) !\)
Ans: Given that, we have \((n+1) !=12 \times(n-1) !\)
We know that, factorial is, 
\(n !=n \times(n-1) \times(n-2) \times(n-3) \times \ldots \times 3 \times 2 \times 1\)
\(\therefore(n+1) \times n \times(n-1) !=12 \times(n-1) !\)
\(\Rightarrow n(n+1)=12\)
\(\Rightarrow(n+1) n=4 \times 3 \quad\) [We can write \(12\) as \(4 \times 3\)]
On comparing both sides, we have, \(n=3\)
Hence, the required value of \(n\) is \(3\).

Q.5. Find the value of \(n\), if \(\frac{n !}{2 !(n-2) !}\) and \(\frac{n !}{4 !(n-4) !}\) are in the ratio \(2: 1\)
Ans: Given that, we have \(\frac{n !}{2 !(n-2) !}: \frac{n !}{4 !(n-4) !}=2: 1\)
We know that, factorial is, 
\(n !=n \times(n-1) \times(n-2) \times(n-3) \times \ldots \times 3 \times 2 \times 1\)
\(\therefore \frac{n !}{2 !(n-2) !} \times \frac{4 !(n-4) !}{n !}=\frac{2}{1}\)
\(\Rightarrow \frac{4 !(n-4) !}{2 !(n-2) \times(n-3) \times(n-4) !}=\frac{2}{1}\)
\(\Rightarrow \frac{4 \times 3 \times 2 !}{2 !(n-2)(n-3)}=\frac{2}{1}\)
\(\Rightarrow(n-2)(n-3)=6\)
\(\Rightarrow(n-2)(n-3)=3 \times 2 \quad[\because 6=3 \times 2]\)
On comparing both sides, we have, \(n-2=3\) and \(n-3=2\)
\(\Rightarrow n=5\)
Hence, the required value of \(n\) is \(5\).

Q.6. Prove that: \(\frac{{(2n)!}}{{n!}} = \{ 1 \cdot 3 \cdot 5 \ldots (2n – 1)\} {2^n}\)
Ans: Given: \(\frac{{(2n)!}}{{n!}} = \{ 1 \cdot 3 \cdot 5 \ldots (2n – 1)\} {2^n}\)
\(\Rightarrow \frac{(2 n) !}{n !}=\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \ldots(2 n-2)(2 n-1)(2 n)}{n !}\)
\( = \frac{{\{ 1 \cdot 3 \cdot 5 \cdot 7 \ldots (2n – 1)\}  \cdot \{ 2 \cdot 4 \cdot 6 \cdot 8 \ldots (2n – 2)(2n)\} }}{{n!}}\)
\( = \frac{{\{ 1 \cdot 3 \cdot 5 \cdot 7 \ldots (2n – 1)\} {2^n}\{ 1 \cdot 2 \cdot 3 \cdot 4 \ldots (n – 1)n\} }}{{n!}}\)
\( = \frac{{\{ 1 \cdot 3 \cdot 5 \cdot 7 \ldots (2n – 1)\}  \cdot {2^n} \cdot n!}}{{n!}}\)
\( = \{ 1 \cdot 3 \cdot 5 \cdot 7 \ldots (2n – 1)\} {2^n}\)
Hence, proved.

Q.7. Find the value of \(n\), if \((n+2) !=60[(n-1) !]\)
Ans: Given that, we have \((n+2) !=60[(n-1) !]\)
As we know that, the formula for the factorial is \(n !=n \times(n-1) \times(n-2) \times(n-3) \times \ldots \times 3 \times 2 \times 1\)
Therefore, \((n+2)(n+1)(n)(n-1) !=60[(n-1) !]\)
\(\Rightarrow(n+2)(n+1)(n)=60\)
\(\Rightarrow(n+2)(n+1)(n)=5 \times 4 \times 3\) [We can write \(60\) as \(5 \times 4 \times 3\)]
On comparing both sides, we have, \(n=3\)
Hence, the required value of \(n\) is \(3\).
Hence, proved.

Q.8. Find the value of \(n\), if \((n+1) !=90[(n-1) !]\)
Ans: Given that, we have \((n+1) !=90[(n-1) !]\)
As we know that, the formula for the factorial is
\(n !=n \times(n-1) \times(n-2) \times(n-3) \times \ldots \times 3 \times 2 \times 1\)
Therefore, \(n+1(n)(n-1) !=90(n-1) !\)
\(\Rightarrow(n+1) n=90\)
\(\Rightarrow(n+1) n=10 \times 9\) [We can write \(90\) as \(10 \times 9\)]
On comparing both sides, we have, \(n=9\)
Hence, the required value of \(n\) is \(9\).

Summary

Factorial of a number is the product of all positive integers less than and equal to the given number. It is denoted by the given integer and an exclamation point. The factorial notation can be found for any positive integer. Factorials of negative integers and rational numbers are not defined. The factorial of zero is \(1\).

Frequently Asked Questions (FAQs)

Q.1. How to find factorial notation?
Ans: The factorial notation can be found for any positive integer. The factorial notation of a number \(n\) is denoted by \(n !\) and its formula is given by \(n !=n \times(n-1) \times(n-2) \ldots 3 \times 2 \times 1\).

Q.2. What is a factorial of \(11\)?
Ans: From the definition of factorial we have, \(n !=1 \times 2 \times 3 \times 4 \ldots \times(n-1) \times n\)
To find \(11!\), just substitute \(n=11\) in above formula i.e., \(n !=1 \times 2 \times 3 \times 4 \ldots \times(n-1) \times n\)
Thus, \(11 !=1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11\)
\(=39916800\)

Q.3. How do you write factorial notation?
Ans: The factorial notation is the exclamation mark, and you will see it directly following a number. 
For example \(4!\), we can read it as ‘four factorial’.

Q.4. Where is factorial used?
Ans: In mathematical analysis, factorials are used in power series for the exponential function and other functions. They also can be used in algebra, number theory, probability theory, and computer science.

Q.5. How are factorials useful?
Ans: It’s very useful when we are trying to count how many different orders are for things or in how many different ways we can combine the things. Factorials are used to find the number of patterns, solve permutations and combinations problems, find out the probability of event problems, etc.

Q.6. What does factorial mean in real life?
Ans: The factorial of n items gives you the number of ways you can arrange the given items.
For example: If there are two coins, we can arrange them in two different ways.

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