Factorial Notation

Factorial notation is used to find the factorial value of any positive natural number. The factorial notation of a natural number n is n!. The factorial of n is represented as n! = 1 x 2 x 3 ....(n - 2) x (n - 1) x n. The factorial notation is prominently used in the formulas of permutation and combination.

Let us learn more about factorial notation, formulas of factorial notation, application of factorial notation, with the help of examples, FAQs.

Factorial notation refers to a symbol '!'. Symbolically the factorial of a natural number n is written as n!. In factorial notation, the factorial of a natural number is equal to the product of all the natural numbers in sequence from 1 to n. For example, the factorial of 5 is written as 5! and is equal to 5 x 4 x 3 x 2 x 1. Further, let us try to understand the history and the reasoning of the concept of factorial notation.

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

The factorial of a natural number only uses the operation of multiplication across the sequence of natural numbers. Let us check the factorial of a set of natural numbers.

• 0! = 1
• 1! = 1
• 2! = 2 x 1 = 2
• 3! = 3 x 2 x 1 = 6
• 4! = 4 x 3 x 2 x 1 = 24
• 5! = 5 x 4 x 3 x 2 x 1 = 120
• 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

The factorial of a number can be easily calculated by taking the product of successive positive numbers from one to the number, for which we need to find the factorial. The factorial notation is a function which multiplies every number with the numbers below it. So, n factorial is the product of the first n natural numbers and is represented as n!

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

Here we write n! = 1 x 2 x 3 x 4 x ......n or n! = n(n - 1)(n - 2)(n - 3) .....3 . 2. 1. Factorial notation writes the factorial of any positive natural number and is equal to the successive natural numbers from n and in a reducing manner up to 1.

The factorial of a number n is equal to the product of the number and the factorial of a number one less than the given number.

n! = n x (n - 1)!

The factorial notation is prominently used in formulas of permutations and combinations. Permutations refer to the arrangement of r things from the given n number of things. Combinations refer to the number of subgroups containing r things each, which can be formed from the given number of n things. Both permutations and combinations use factorial notation.

The permutations refer to the different possible arrangements which can be formed from r things, taken from n things. The permutation of r things taken from n things is equal to the product of the factorial of n, divided by the difference of the factorial between n and r.

Permutation: \(^nP_r = \dfrac{n!}{(n - r)!}\)

The combinations refer to the different possible groups which can be formed from r things taken from n things. The combination of number of groups that can be formed and containing r things in each group, taken from n things is equal to the factorial of n divided by the product of the factorial of r, and the factorial of the difference of n and r.

Combination: |(^nC_r = \dfrac{n!}{r!.(n - r)!}\)

☛Related Topics

The following topics help for a better understanding of factorial notation.

• Permutation and Combination
• Probability Density Function
• Theorem Of Total Probability
• Multiplication Rule of Probability

What Is Factorial Notation?

The factorial notation is taken for natural numbers and is represented using the symbol '!'. The factorial notation for a natural number n is written as n!. The factorial of n is n! = 1 x 2 x 3 x 4 ..... (n - 2) x (n - 1) x n.

How to Find Factorial Notation?

The factorial notation can be found for any positive natural number. The factorial notation of a number n is n! and n! = n x (n - 1) x (n - 2) ....3 x 2 x 1. As an example, the factorial of 4 is 4! = 4 x 3 x 2 x 1 = 24.

What Is The Formula For Factorial Notation?

The formula for factorial notation for a positive natural number n is n! = n x (n - 1) x (n - 2) ....3 x 2 x 1.

What Is The Use Of Factorial Notation?

The factorial notation is used prominently in the formulas of permutation and combination. The formula of permutation is \(^nP_r = \dfrac{}{}\), and the formula of combination is \(^nC_r = \dfrac{n!}{r!.(n - r)!}\).

How To Calculate 0! Using Factorial Notation?

The value of 0! is calculated using the same factorial notation. The value of 0! is equal to 1. 0! = 1