Factorial

Factorial is a fundamental concept in combinatorics as factorials play important roles in various mathematical formulas such as permutations, combinations, probability, and many other formulas. Factorial of any natural number “n” is defined as the product of all natural numbers till n.

In this article, we’ll delve into the intricacies of factorials, exploring factorial notation, the diverse range of factorial formulas, and techniques for computing factorials. Additionally, we’ll touch upon the properties and practical applications of factorials, provide illustrative examples, and address common questions pertaining to this topic. Let’s embark on our journey of understanding factorials.

What is Factorial?

Factorial is the product of n numbers until it reaches up to 1. It we want to calculate the factorial of n, then we multiply the number less than or equal to n until it encounters 1. In other words, the multiplication of 1 to n is called the factorial of n.

The factorial of the number n can be also defined as the product of the number n and the factorial (n -1).

Factorial Notation

The notation of the factorial is “!” or “⌋”. If we have to find the factorial of the number n then, it is written as n! or n⌋.

Let’s consider some examples of factorials:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 4! = 4 × 3 × 2 × 1 = 24
• 3! = 3 × 2 × 1 = 6
• 2! = 2 × 1 = 2
• 1! = 1

Factorial of 0

As a factorial is defined as the product of natural numbers up to the number under consideration, but in the case of 0, if we were to follow the same definition, it would result in 0. However, this would lead to inconsistencies with many already proven results. Therefore, factorial is initially defined in such a way that the factorial of 0 is 1. This definition makes sense on a larger scale, and we have further demonstrated its validity. As we generalize factorials into gamma functions, the result remains the same.

Thus, the factorial of 0 is defined as 1 and is represented as 0!

Factorial Formula

The factorial formula is the formula in which we multiply all the number less than n until it is equal to 1. The factorial formula is given by:

n! = n × (n -1) × (n – 2) … 3 × 2 × 1

OR

n! = i

OR

n! = n× (n – 1)!

How to Find Factorial of a Number?

To find the factorial of a number we apply following steps:

• First, check if the given number whose factorial is to be evaluated is positive or negative.
• If the number is negative the factorial of negative number is undefined.
• If the number is positive, find the factorial of the number using the above factorial formulas.

Factorial Examples

As we can calculate the factorials for any non-negative numbers, thus there can be infinitely many examples of factorials. Let’s consider some of those examples as follows:

Factorial of 5

The Factorial of 5 is obtained by multiplying numbers from 1 to 5.

Factorial of 5 = 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorial of 10

The Factorial of 10 is obtained by multiplying numbers from 1 to 10.

Factorial of 10 = 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800

Factorial of 100

The Factorial of 100 is obtained by multiplying numbers from 1 to 100.

Factorial of 100 = 100! = 100 × 99 × 98 × 97 × 96 × . . . × 5 × 4 × 3 × 2 × 1 = 9.33262154 × 10¹⁵⁷

Properties of Factorial

Some of the properties of factorial are:

• For any non-negative integer n, 
  • n! = n × (n – 1) × (n – 2) × … × 3 × 2 × 1
• Factorial can be defined recursively as follows:
  • n! = n(n – 1)! [ Where  0! = 1]
• 0! is defined to be 1 by convention.
• For any non-negative integer n, n! is always an integer.
• As ∏ is used to represent product of terms in sequence, thus factorial of n can also be represented as:
  • n! = ∏(i = 1 to n) i.
• Factorial of negative numbers are undefined.
• The factorial of large numbers can grow very rapidly. For example, 10! = 3,628,800, 15! = 1,307,674,368,000, and so on. 

Factorials 1 to 20

The following table list of first 20 factorials, from 1 to 20.

Number	Factorial
1	1
2	2
3	6
4	24
5	120
6	720
7	5040
8	40320
9	362880
10	3628800
11	39916800
12	479001600
13	6227020800
14	87178291200
15	1307674368000
16	20922789888000
17	355687428096000
18	6402373705728000
19	121645100408832000
20	2432902008176640000

Applications of Factorials

There are various applications of the factorial. Some of the applications of factorials are listed below:

• Factorial is used in permutations.
• Factorials are used in combinations.
• It is used in probability formulas.
• It is used in binomial expansion.

Factorials in Combinatorics

In calculation of both permutation and combination is used as the formula for both involves the factorials. Let’s see formula for permutation and combination as well as their example as well.

Permutation Formula

The formula for calculating permutations, denoted as ⁿP_r  which represents the number of ways to arrange r objects from a set of n distinct objects without repetition and formula for permutation is given by:

ⁿP_r = n! / (n – r)!

Where,

• n is the total number of distinct objects to choose from,
• r is the number of objects to be chosen and arranged,
• n! is the product of all positive integers from 1 to n,
• (n – r)! is the product of all positive integers from 1 to (n – r).

Learn More about Permutation.

Example: Evaluate the value of ⁵P₃.

Solution:

By permutation formula

ⁿP_r = n! / (n – r)!

⇒ ⁵P₃ = 5! / (5 – 3)!

⇒ ⁵P₃ = 5! / 2!

⇒ ⁵P₃ = 120 / 2

⇒ ⁵P₃ = 60

Combination Formula

The formula for calculating combinations, denoted as ⁿC_r, where n is the total number of items to choose from, and r is the number of items to choose without replacement. This formula is given as follows:

ⁿC_r = n! / [r! × (n – r)!]

Where,

• n is the total number of distinct objects to choose from,
• r is the number of objects to be chosen and arranged,
• n! is the product of all positive integers from 1 to n,
• (n – r)! is the product of all positive integers from 1 to (n – r).

Read more Combination.

Example: Find the value of ⁴C₂.

Solution:

By combination formula

ⁿC_r = n! / [r! × (n – r)!]

⇒ ⁴C₂ = 4! / [2! × (4 – 2)!]

⇒ ⁴C₂ = 4! / [2! × 2!]

⇒ ⁴C₂ = 24 / [2 × 2]

⇒ ⁴C₂ = 24 / 4

⇒ ⁴C₂ = 12

Factorials in Probability

Factorials are used in multiple formulas in probability, as factorials help us calculate the number of ways of things with the help of principle of counting, permutation, and combination. Let’s consider an example of Probability where we calculate the probability of any event with the help of factorials.

Example: A box contains different colored balls. There is 15% chance of getting a red ball. What is the probability that exactly 4 balls are red out of 10.

Solution:

Applying binomial distribution

P(X = r) = ⁿC_r p^r q^n-r

n = 10, p = 0.15, q = 0.85, r = 4

⇒ P(X = 4) = ¹⁰C₄ (0.15)⁴ (0.85)⁶

⇒ P(X = 4) = [10! / {4! × 6!}] (0.15)⁴ (0.85)⁶

⇒ P(X =4) = [{10× 9 × 8 × 7} / 24] (0.15)⁴ (0.85)⁶

⇒ P(X = 4) = 0.04

Solved Examples on Factorial

Example 1: Evaluate the following.

• Factorial of 1
• Factorial of 3
• Factorial of 4
• Factorial of 6
• Factorial of 7
• Factorial of 8
• Factorial of 9

Solution:

Factorial of 1 = 1! = 1

Factorial of 3 = 3! = 3 × 2 × 1 = 6

Factorial of 4 = 4! = 4 × 3 × 2 × 1 = 24

Factorial of 6 = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Factorial of 7 = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Factorial of 8 = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320

Factorial of 9 = 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 =362880

Example 2: What is the value of factorial: 14! / (11! × 4!)

Solution:

14! / (11! × 4!) = (14 × 13 × 12 × 11!) / (11! × 4!)

⇒ 14! / (11! × 4!) = (14 × 13 × 12) / 4!

⇒ 14! / (11! × 4!) = (14 × 13 × 12) / (4 × 3 × 2 × 1!)

⇒ 14! / (11! × 4!) = (14 × 13 × 12) / (12 × 2 )

⇒ 14! / (11! × 4!) = (7 × 13)

⇒ 14! / (11! × 4!) = 91

Example 3: Evaluate the expression 6! – 3!

Solution:

6! – 3! = (6 × 5 × 4 × 3!) – 3!

⇒ 6! – 3! = (6 × 5 × 4 × 3!) – 3!

⇒ 6! – 3! = (120 × 3!) – 3!

⇒ 6! – 3! = 3![120 – 1]

⇒ 6! – 3! = 6 × 119

⇒ 6! – 3! = 714

Example 4: If (1 / 6!) = (x / 8!) – (1 / 7!), then find the value of x.

Solution:

(1 / 6!) = (x / 8!) – (1 / 7!)

⇒ (1 / 6!) = (x / 8 × 7!) – (1 / 7!)

⇒ (1 / 6!) = (1 / 7!)[(x / 8) – 1]

⇒ (1 / 6!) = {1 / (7 ×6!)}[(x / 8) – 1]

⇒ (1 / 6!) = (1 / 6!)(1 / 7 )[(x / 8) – 1]

⇒ 1 = (1 / 7 )[(x / 8) – 1]

⇒ 7 = (x / 8) – 1

⇒ (x / 8) = 7 + 1

⇒ (x / 8) = 8

⇒ x = 64

Example 5: How many 4-digit numbers can be formed using the digits 4,6,7,9 in each of which no digit is repeated?

Solution:

Given:

Digits: 4, 6, 7, and 9

Number of digits = 4

We have to arrange these digits to form a 4-digit number.

The number of ways for arranging these digits to form a 4-digit number is 4!

and 4! = 4 × 3 × 2 × 1 = 24

Thus, there are 24 ways in which a 4 digit number can be formed without repeating the digits.

Example 6: Evaluate the expression 3! (2! × 0!)

Solution:

3! (2! × 0!) = (3 × 2 × 1) (2 × 1 × 1) [By using factorial formula and 0! = 1]

⇒ 3! (2! × 0!) = 6 × 2

⇒ 3! (2! × 0!) = 12

Practice Problems on Factorials

Problem 1: Evaluate.

• (8! × 7!) / 6!
• 7! / 4!
• 10! – 9!

Problem 2: Simplify.

• (7 + 3)! / 2!
• 6! / (4! × 2!)
• (9!) / [(7!) × (2!)]
• (6!) / [(5!) × (3!)]
• (12!) / [(11!) × (10!)]

Problem 3: Find the Value of n if

• n! = 120
• (n – 1)! = 24
• (n + 2)! = 720
• (n – 2)! = 120

Problem 4: Find the factorial of 9 and subtract the factorial of 6.

FAQs on Factorial

1. What is Factorial in Math?

Factorial of a number is the product of numbers less than n up to 1.

2. What is the Formula for the Factorial of any Number n?

The formula for the factorial is given by:

n! = n × (n -1) × (n – 2) … 3 × 2 × 1

3. How is a Factorial Calculated?

To calculate factorial of any number n i.e., n!, multiply all integers from 1 to n together. For example, 3! = 3 × 2 × 1 = 6.

3. What is the Value of 0!?

The value of factorial zero i.e., 0! = 1.

4. What is the Notation of Factorial?

The notation of factorial is !

5. Why Factorial is Used?

The factorial is used in permutations, combinations, binomial theorem, probability etc.

6. What is the Purpose of Factorials?

The purpose of factorials is to represent the product of all positive integers from 1 to a given number, commonly used in combinatorics and mathematical calculations.

7. Can Factorials be Calculated for Non-Integer Values?

Factorials are defined for non-negative integers only, while the gamma function, an extension of factorials, is defined for all non-integer values.

8. What is Factorial of 5?

Factorial of 5 is denoted by 5! or 5⌋ and is equal to 120.

9. What is the value of 6 Factorial?

The value of factorial is 6! = 720

10. What is Factorial of 7?

Factorial of 7 is denoted by 7! and is equal to 5040.

10. What is factorial of 100?

Factorial of 100 is 9.33262154 × 10¹⁵⁷.

11. What is the value of 4 Factorial?

The value of 4 Factorial is 4! = 120