Factorial in Maths: Definition, Formulas & Applications (2024)

The factorial of any natural number "n" is defined as the product of all natural numbers till n. It plays an important role in various mathematical concepts such as permutations, combinations, probability, and many others.

n! = n \times (n-1) \times (n-2) \times \cdots \times 1

The notation of the factorial function is "!" or "⌋". If we have to find the factorial of the number n then, it is written as n! or n⌋. Let's understand it with some examples:

• 0! = 1 ( Value of Factorial 0 is 1 because it shows the number of possible ways to arrange a data set with no value in it is 1)
• 1! = 1
• 2! = 2 × 1 = 2
• 3! = 3 × 2 × 1 = 6
• 4! = 4 × 3 × 2 × 1 = 24
• 5! = 5 × 4 × 3 × 2 × 1 = 120

Why is 0! (Factorial of 0) equal to 1?

When you're counting groups of things or arranging objects, multiplying by 1 doesn’t change the total number. In math, the factorial of 0 is defined as 1 to keep the pattern consistent with larger numbers. It's like saying, "If you have nothing to arrange, there's only one way to do nothing." So, we assign 0! the value of 1 to make the math rules work smoothly.

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

Factorial Formula

The factorial formula can be expressed as:

n! = n × (n -1) × (n - 2) ... 3 × 2 × 1

Read in Detail: Factorial Formula

Calculating Factorial

Calculating factorials is a fundamental operation in mathematics, especially in combinatorics. The factorial of a non-negative integer n is denoted as n! and is defined as the product of all positive integers less than or equal to n.

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

List of Factorials of Numbers 1 to 15

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.

Applications of Factorial

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

1. Combinatorics: Factorials are essential in combinatorics, which is the study of counting, arrangement, and combination of objects. They are used to calculate:

• Permutations
• Combinations

2. Probability: In probability, factorials are used to determine the number of possible outcomes in experiments. For example:

• The probability of drawing a specific hand of cards from a deck can be calculated using combinations, which involve factorials.
• Factorials are also used in calculating probabilities in binomial distributions, where the number of ways to achieve a certain number of successes in trials is calculated using factorials.

3. Statistics: Factorials are used in various statistical formulas, including:

• Binomial Coefficient
• Poisson Distribution

4. Mathematical Series: Factorials are utilized in the expansion of power series, such as the Taylor and Maclaurin series. These series represent functions as infinite sums of terms calculated from the derivatives of functions at a single point, where factorials appear in the denominators.

5. Games and Puzzles: Factorials are used in games that involve arranging items or characters in specific orders, such as in board games and card games. The number of possible arrangements can often be calculated using factorials.

6. Computer Programming: Factorials are commonly implemented in programming for various applications such as:

• Generating permutations and combinations.
• Solving mathematical problems that require combinatorial logic.

Factorials in Combinatorics (Permutations And Combinations)

In calculation of both permutation and combination is used as the formula for both involves the factorials. Let's see Permutation Formula and Combination Formula along with their examples.

Read More: Permutation and Combination

Example 1. 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

See Also

Factorial CalculatorFactorial Function !What is a Factorial? How to Calculate Factorials with Examples

Example 2. 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

Factorial in Statistics and 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

Also, Check:

• Number System
• Binomial Theorem

Solved Questions on Factorial Function

Que 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

Que 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

Que 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

Que 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

Que 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.

Que 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

Factorial Practice Questions with Answer Key

You can download the practice questions from below with answers:

Conclusion

The factorial function is a fundamental concept in mathematics with wide-ranging applications in permutation and combination, probability, and algebra. By multiplying a positive integer by all positive integers below it, factorials help us solve problems related to permutations and combinations, providing essential tools for analyzing and organizing data. Understanding factorials not only aids in mathematical computations but also enhances problem-solving skills in various fields.

FAQs on Factorial in Maths

What is Factorial?

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

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

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.

What is the Value of 0!?

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

What is the Notation of Factorial function?

The notation of factorial functionis !

Why Factorial function is Used?

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

What is the Purpose of Factorial function?

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

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.

What is Factorial of 5?

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

What is the value of 6 Factorial?

The value of factorial is 6! = 720

What is Factorial of 7?

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

What is factorial of 100?

Factorial of 100 is 9.33262154 × 10¹⁵⁷.

What is the value of 4 Factorial?

The value of 4 Factorial is 4! = 120