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(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.

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.

The formula for n factorial is: \[n! = n \times (n - 1)!\]

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:

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.

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:

Combination is a grouping of outcomes in which the order does not matter. It can be calculated with the formula:

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.

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:

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,  ⁿP_r = \(\dfrac{{n!}}{{(n - r)!}}\) & ⁿC_r = \(\dfrac{{n!}}{{r!(n - r)!}}\) respectively.