# Factorial

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. In short, a factorial is a function, that multiplies a number by every number below it. In this article, let us discuss the definition of the factorial, its notation, formula, examples and so on in detail.

Also, Check: Factorial Calculator

## 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, let’s discuss the factorial definition, formula and examples.

## Factorial Notation

The multiplication of all positive integers says “n”, 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 “n!”.

## Factorial Formula

The formula to find the factorial of a number is

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

For an integer n ≥ 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 factorial number and factorial of that number minus 1. It is given by:

n! = n. (n-1) !

For example, the factorial of 10 is written as

10! = 10. 9 !

10! = 10 (9 × 8 × 7 × 6 × 5× 4 × 3 × 2 × 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 “n” possible distinct objects.

## Factorials of Numbers 1 to 10 Table

The list of factorial values from 1 to 10 are:

 Factorial of a Number Expansion Value 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 6! 6× 5 ×4 ×3× 2× 1 720 7! 7× 6× 5 ×4× 3× 2 ×1 5,040 8! 8× 7× 6× 5× 4× 3 ×2× 1 40,320 9! 9 ×8× 7× 6× 5× 4 ×3× 2 ×1 362,880 10! 10× 9×8× 7× 6× 5×4× 3× 2× 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 Examples

Example 1:

What is the factorial of 5?

Solution:

We know that the factorial formula is

n! = n × (n-1)×(n-2)×(n-3)× ….× 3×2×1

So the factorial of 5 is

5! = 5× (5-1)× (5-2)× (5-3)× 1

5! = 5 ×4 ×3 ×2 ×1

5! = 120

Therefore, the factorial of 5 is 120

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.

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