## Factorial of a number N

Enter any non-negative number between 1 to 100 and press 'Calculate'.

#### Factorial Calculator

Result:

 Factorial: 0

### What is Factorial?

Factorial of a number is the product of all the numbers less than that number. In other words, the factorial function gives the product of a number with every number less than it. It is denoted by the exclamation point, “!”.

For example, 4! = 4 × 3 × 2 × 1 = 24. The factorial can only be calculated for positive integers.

For an arbitrary integer n, the n! = n × (n-1) × (n-2) × (n-3) × … 2 × 1. The factorial of 0 is 1.

Some examples are as follows.

### Properties of Factorial

Given below are some important properties of factorial.

Number – The number for which the factorial is to be calculated is always a positive integer.

Factorial – The factorial for the number is the product of all the numbers less than that number.

Double factorial – The double factorial for a number is the product of all the numbers less than that number, and having the same parity as that number. It is defined differently for even and odd numbers.

### Factorial formula

For any positive integer n, the factorial is defined as

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

For example, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

The factorial representation using the product notation is $$n! = \prod_{k=1}^n {k}$$

So, 6! can be written as: $$6! = \prod_{k=1}^6 {k}$$

The factorial for a number n can also be written in the form of a recurrence relation. This relation defines the factorial as the product of the number n with the factorial of the number less than n by one. This is written as n! = n × (n-1)! For example, 6! = 6 × (6-1)! = 6 × 5! = 6 × 120 = 720.

The double factorial formula is defined as follows.

If n is even and n > 0, n!! = n × (n-2) × (n-4) × (n-6) × … × 4 × 2.

If n is odd and n > 0, n!! = n × (n-2) × (n-4) × (n-6) × … × 3 × 1.

If n = 0, 0!! = 1.

For example, 6!! = 6 × (6-2) × (6-4) = 6 × 4 × 2 = 84, and 7!! = 7 × (7-2) × (7-4) × (7-6) = 7 × 5 × 3 × 1 = 105.

### Characteristics of factorial

Given below are some characteristics of factorial.

• The factorial is defined only for positive integers.
• For any positive integer n, n! = n × (n-1)!, or $${n! \over n} = (n-1)!$$
• The factorial of 0 is 1, that is 0! = 1.
• The factorial of 1 is also 1. 1! = 1.
• Double factorial is a kind of multiple factorial denoted by n!!. It is the factorial of all the numbers less than n and having the same parity as n. There are two different formulas for double factorial depending on whether n is even or odd.

• ### Areas of application

The factorial function has quite a few applications in mathematics as well as real-life.

1) Factorials are immensely useful in permutations and combinations when we need to find the different orders of arrangement of a set of objects, or the various ways a set of objects can be combined.
The factorial n! of a number n, shows how many different ways we can arrange n objects. For arranging the first object, we have n choices. For each n choice, we obtain n-1 choices for the second object. So, for arranging the first two objects in order, we have n(n-1) choices. Now, for the third object, we have n-2 choices. This continues till we are left with 1 choice for the nth object.
Hence, n! = n × (n-1) × (n-2) × (n-3) × … × 3 × 2 × 1 illustrates how many ways we can arrange n objects in order. For example, if you have 3 books, there are 6 different ways to arrange them on a bookshelf. If you have 5 books, there are 120 different ways to arrange them.

2) The factorial function is also used when we need to choose k objects from a set of n objects. This is the definition of combinations, given by n!/(k!(n-k)!), which is denoted as nCk. For example, if a college offers 5 courses, and a student is permitted to choose only 3 among them, the number of ways they can do so is, 5C3 = 5!/(3!2!) = 120/(6*2) = 120/12 = 10. Hence, the student can choose 3 courses out of 5 in 10 different ways.

3) The factorial function is frequently seen in calculus. The Taylor’s theorem is one such example. $$f(x) = \sum_{k=0}^{\infty} {{{f^{(n)}(x_0)} \over k!} (x-x_0)^k}$$

4) It is also used to define trigonometric functions like sine and cosine, and also the exponential function.

$$e^x = 1+{x\over1!}+{x^2\over2!}+{x^3\over3!}+... \; -\infty < x < \infty$$

$$sin \; x = x - {x^3\over3!}+{x^5\over5!}-{x^7\over7!}+{x^9\over9!}-... \; -\infty < x < \infty$$

$$cos \; x = 1 - {x^2\over2!}+{x^4\over4!}-{x^6\over6!}+{x^8\over8!}-... \; -\infty < x < \infty$$

5) Other than positive integers, factorials can also be calculated for real and complex numbers. This can be done using gamma functions.