# Factorial Formula

Before going to learn the factorial formula let us recall what is a factorial. We can understand this by taking some examples. "5 factorial" is written as 5! and its value is 5! = 5 × 4 × 3 × 2 × 1. In the same way "8 factorial" is written as 8! and its value is 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1. Thus, we write "n factorial" as n! (as shown below). By this time you must have understood the definition of factorial. Let us learn more about the factorial formula along with few examples.

## What Is Factorial Formula?

The factorial formula helps in calculating the factorial of any given number. A factorial of a number is the function that finds the product of all natural numbers less than or equal to the given number. Symbolically the factorial can be represented as "!". For example, 6! = 1 × 2 × 3 × 4 × 5 × 6 = 720. Factorials are commonly used is in permutations & combinations. Here are the factorial formulas.

**Note: **Note that 0! is always 1. Here you can see how 0! is 1.

### Factorial Formula 1

By the definition of factorial, the factorial of a number n is the product of all natural numbers(taken in decreasing order) from n to 1. i.e.,

n! = n × (n - 1) × (n - 2) × .... × 1

### Factorial Formula 2

We know that

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

From the above two equations, we can write n! as n times (n - 1)!. i.e.,

n! = n (n - 1)!

This formula is widely used in problems for making calculations easier.

## Factorials Examples

The table below shows the factorials of some numbers using both of the above formulas.

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

1 | 1 | 1 | 1 |

2 | 2 × 1 | = 2 × 1! | = 2 |

3 | 3 × 2 × 1 | = 3 × 2! | = 6 |

4 | 4 × 3 × 2 × 1 | = 4 × 3! | = 24 |

5 | 5 × 4 × 3 × 2 × 1 | = 5 × 4! | = 120 |

We can see the applications of the factorial formula in the following section.

## Examples Using Factorial Formula

**Example 1: **Find the values of the following factorials. a) 8! b) 10!

**Solution:**

Using the factorial formula, n! = n × (n - 1) × (n - 2) × .... × 1.

We will use this formula to compute the given factorials.

a) 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320

b) 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

**Answer: **a) 8! = 40,320 b) 10! = 3,628,800

**Example 2: **Determine: 5! (7 - 3)!.

**Solution:**

Using the Factorial formula,

5! (7 - 3)! = 5! × 4!

= (5 × 4 × 3 × 2 × 1) × (4 × 3 × 2 × 1)

= 120 × 24

= 2880

**Answer: **5! (7 - 3)! = 2880.

**Example 3: **Simplify: 11! / [5! × 3!].

**Solution:**

We will find all factorials using the factorial formula and simplify.

11! / [5! × 3!] = [11 × 10 × 9 × 8 × 7 × 6 × 5!] / [5! × 3!]

= [11 × 10 × 9 × 8 × 7 × 6] / 3!

= [11 × 10 × 9 × 8 × 7 × 6] / [3 × 2 × 1]

= [11 × 10 × 9 × 8 × 7 × 6] / 6

= 11 × 10 × 9 × 8 × 7

= 55440

**Answer:** 11! / [5! × 3!] = 55440.

## FAQs Factorial Formula

### What Is Factorial Formula?

The factorial formula says the factorial of a number is the product of the number and all the natural numbers from n to 1 in the decreasing order. The factorial formulas are:

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

### What Are the Applications of Factorial Formula?

We use the factorial to find the number of ways in which a given number of things can be arranged among themselves. For example, 4 things can be arranged among themselves in 4! = 24 ways. We use factorials in permutations and combinations as well.

### How To Derive 0! = 1 Using Factorial Formula?

0! is 1 and it can be derived as follows:

**Step 1:**4! = 4 × 3 × 2 × 1 = 24**Step 2:**Dividing the above equation by 4, we get, 3! = 3 × 2 × 1 = 6**Step 3:**Dividing the above equation by 3, we get, 2! = 2 × 1 = 2**Step 4:**Dividing the above equation by 2, we get, 1! = 1**Step 5:**Dividing the above equation by 1, we get, 0! = 1

### Can Factorial Formula be Applied For a Negative Number?

No, the factorial formula cannot be applied for negative numbers. It is defined only for whole numbers.