In Mathematics, factorial is an important function, which is used to find how many ways things can be arranged or the ordered set of numbers. The well known interpolating function of the factorial function was discovered by Daniel Bernoulli. The factorial concept is used in many mathematical concepts such as probability, permutations and combinations, sequences and series, etc. In short, a factorial is a function that multiplies a number by every number below it till 1. For example, the factorial of 3 represents the multiplication of numbers 3, 2, 1, i.e. 3! = 3Β Γ 2Β Γ 1 and is equal to 6.Β In this article, you will learn the mathematical definition of the factorial, its notation, formula, examples and so on in detail.
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What is Factorial?
In Mathematics, factorial is a simple thing. Factorials are just products. An exclamation mark indicates the factorial. Factorial is a multiplication operation of natural numbers with all the natural numbers that are less than it. In this article, letβs discuss the factorial definition, formula and examples.
Factorial Notation
The multiplication of all positive integers, say βnβ, that will be smaller than or equivalent to n is known as the factorial. The factorial of a positive integer is represented by the symbol βn!β.
Factorial Formula
The formula to find the factorial of a number is
n! = n Γ (n-1) Γ (n-2) Γ (n-3) Γ β¦.Γ 3 Γ 2 Γ 1
For an integer n β₯ 1, the factorial representation in terms of pi product notation is:
\(n! = \prod_{i=1}^{n}i\)
From the above formulas, the recurrence relation for the factorial of a number is defined as the product of the factorial number and factorial of that number minus 1. It is given by:
n! = n. (n-1) !
Factorial of a Number
To find the factorial of any given number, substitute the value for n in the above given formula. The expansion of the formula gives the numbers to be multiplied together to get the factorial of the number.
Factorial of 10
For example, the factorial of 10 is written as
10! = 10. 9 !
10! = 10 (9 Γ 8 Γ 7 Γ 6 Γ 5Γ 4 Γ 3 Γ 2 Γ 1)
10! = 10 (362,880)
10! = 3,628,800
Therefore, the value of 10 factorial is 3,628,800
The factorial operation is encountered in many areas of Mathematics such as algebra, permutation and combination, and mathematical analysis. Its primary use is to count βnβ possible distinct objects.
For example,Β the number of ways in which 4 persons can be seated in a row can be found using the factorial. That means, the factorial of 4 gives the required number of ways, i.e. 4! = 4 Γ 3 Γ 2 Γ 1 = 24. Hence, 4 persons can be seated in a row in 24 ways.
Factorials of Numbers 1 to 10 Table
The list of factorial values from 1 to 10 are:
n |
Factorial of a Number n! |
Β Β Β Β Β Expansion |
Value |
1 | 1! | 1 | 1 |
2 | 2! | 2 Γ 1 | 2 |
3 | 3! | 3 Γ 2 Γ 1 | 6 |
4 | 4! | 4 Γ 3 Γ 2 Γ 1 | 24 |
5 | 5! | 5 Γ 4 Γ 3 Γ 2 Γ 1 | 120 |
6 | 6! | 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 | 720 |
7 | 7! | 7 Γ 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 | 5,040 |
8 | 8! | 8 Γ 7 Γ 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 | 40,320 |
9 | 9! | 9 Γ 8 Γ 7 Γ 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 | 362,880 |
10 | 10! | 10 Γ 9 Γ8 Γ 7 Γ 6 Γ 5 Γ4 Γ 3 Γ 2 Γ 1 | 3,628,800 |
What is Sub factorial of a Number?
A mathematical term “sub-factorial”, defined by the term “!n”, is defined as the number of rearrangements of n objects. It means that the number of permutations of n objects so that no object stands in its original position. The formula to calculate the sub-factorial of a number is given by:
\(!n = n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}\)
Factorial of 5
Finding the factorial of 5 is quite simple and easy. This can be found using formula and expansion of numbers. This is given below with detailed steps.
We know that,Β
n! = 1 Γ 2 Γ 3 β¦β¦ Γ n
Factorial of 5 can be calculated as:
5! = 1 Γ 2 Γ 3 Γ 4 Γ 5
5! = 120
Therefore, the value of factorial of 5 is 120.
Video Lesson
Exponent of Prime in Factorial
Factorial Examples
Example 1:
What is the factorial of 6?
Solution:
We know that the factorial formula is
n! = n Γ (n – 1) Γ (n – 2) Γ (n – 3) Γ β¦.Γ 3 Γ 2 Γ 1
So the factorial of 6 is
6! = 6 Γ (6 -1) Γ (6 – 2) Γ (6 – 3) ΓΒ (6 – 4) Γ 1
6! = 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ1
6! = 720
Therefore, the factorial of 6 is 720.
Example 2:
What is the factorial of 0?
Solution:
The factorial of 0 is 1
i.e., 0 ! = 1
According to the convention of empty product, the result of multiplying no factors is a nullary product. It means that the convention is equal to the multiplicative identity.
Practice Problems
Practice the problems given below to understand the concept.
- Evaluate 7! – 5!.
- What is the value of 12!/(10! 4!)
- If (1/6!) = (x/8!) – (1/7!), then what is the value of x?
- Is 4! + 5! = 9!?
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