Factorial

In this mini-lesson, we will learn concepts related to factorial. They include n+1 factorial, the factorial of 10, the factorial of 0, combinations, and permutations.

Before we start, let's see how it all began.

In the year 1677, Fabian Stedman, a British author, defined factorials 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!

You will be surprised to know that the study of factorials is at the root of several topics in mathematics, such as number theory, algebra, geometry, probability, statistics, graph theory, and discrete mathematics, etc.

Look at the factorial calculator below.

Just substitute the value of \(n\) to get its factorial value.

 

Thinking about how to calculate the factorial of a number? Let's learn.

Lesson Plan

A factorial of a number is the function which multiples the number by every natural number below it. Symbolically the factorial can be represented as "!"

So, n factorial is the product of first n natural numbers, represented as n!

So n! or "\(n\) factorial" means:

n! = 1. 2. 3…………………………………n

=  Product of the first n positive integers

=  n(n-1)(n-2)…………………….(3)(2)(1)

For example,

4 factorial, that is, 4! can be written as:

4! = \(4 \times 3 \times 2 \times 1\) = 24

Let us look at a few numbers on the table.

n	n!
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
6	etc...	etc...

• To work out for 6! multiply 120 by 6, to get 720
   
• To work out for 7! multiply 720 by 7, to get 5040

The factorial formula or formula for n factorial is given by,

\[n! = n \times (n - 1)!\]

This means that the factorial of any number is that number times the factorial of that number minus 1

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:

What About Negative Integers?

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.

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One area where they are commonly used is in permutations & combinations.

A permutation is an arrangement of outcomes in which the order does matter.

It can be calculated as

nPr = \(\dfrac{{n!}}{{(n - r)!}}\)

A combination is a grouping of outcomes in which the order does not matter.

It can be calculated as

nCr = \(\dfrac{{n!}}{{r!(n - r)!}}\)

Let us understand by taking examples,

Example 1

In a group of 10 people, Rs.200, Rs.100, and Rs.50 prizes are to be given. How many ways can the prizes be distributed?

Solution:

This is permutation because here the order matters — who get Rs.20 or  Rs.10 or Rs.5

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

In a group of 10 people, three Rs.50 prizes are to be given. 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.

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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

\(9! = 9 \times 8! = 9 \times 40,320 = 362,880\)

Look at the factorial calculator below to calculate combination.

Just drag the slider to change the value of n and r and see the value of n factorial and its combination.

A small factorial table:

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

As you can see, it gets big quite drastically.

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

\(\therefore\) The value is 210

Example 2

 

 

Find the value of  \(5!(6 - 3)!\)

Solution

\[\begin{array}{l}
 5!(6{\rm{ }} - {\rm{ }}3)!{\rm{ }} &= {\rm{ }}5! \times 2! \\[0.2cm]
 {\rm{ }} &= (5 \times 4 \times 3 \times 2 \times 1)(2 \times 1) \\[0.2cm]
 {\rm{ }} &= 120 \times 2 \\[0.2cm]
 {\rm{ }} &= 240 \\
 \end{array}\]

Therefore, the value of  \(5!(6 - 3)!\) is 240

\(\therefore\) The value is 240

Example 3

 

 

How many 5-digit 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 5-digit numbers.

The number of ways for doing this can be done using the factorial.

\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]

\(\therefore\)  the required number of 5-digit numbers is, 120

Example 4

 

 

How many ways can sprinters come first, second, and third in a row with 10 volunteers?

Solution

We need to count all the possibilities in which

we can select 3 different runners in different orders. This can be selected in \(^{{\rm{1}}0}{{\rm{P}}_{\rm{3}}}\) ways.

\[^{{\rm{1}}0}{{\rm{P}}_{\rm{3}}} = \dfrac{{10!}}{{7!}} = 720\]

\(\therefore\) Sprinters can come first, second, and third in 720 ways.

Example 5

 

 

In how many ways can eight people line up from left to right for a group photo?

Solution

 \(\begin{align}&\text{Number of ways 8 people line up}\\[0.2cm]
&= 8! \\[0.2cm]
&= 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\[0.2cm]
&= 40,320 \\
 \end{align}\)

\(\therefore\) Eight people can line up in 40,320 ways.

 

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Let's Summarize

The mini-lesson targeted the fascinating concept of factorial. The math journey around factorial starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

Now you will be able to easily solve problems on n+1 factorial, the factorial of 10, the factorial of 0, combinations, and permutations.

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