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# Factorial

Factorial of a whole number ‘n’ is defined as the product of that number with every whole number till 1. For example, the factorial of 4 is 4×3×2×1, which is equal to 24. It is represented using the symbol ‘!’ So, 24 is the value of 4! 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 the number theory, algebra, geometry, probability, statistics, graph theory, and discrete mathematics, etc.

Thinking about how to calculate the factorial of a number? Let’s learn.

**What Is Factorial?**

The factorial of a number is the function that multiplies the number by every natural number below it. Symbolically, factorial can be represented as “!”. So, *n* factorial is the product of the first *n* natural numbers and is represented as n!

So n! or “n factorial” means: n! = 1. 2. 3…………………………………n = Product of the first n positive integers = n(n-1)(n-2)…………………….(3)(2)(1)

**For example,** 4 factorial, that is, 4! can be written as: 4! = 4×3×2×14×3×2×1 = 24.

Observe the numbers and their factorial values given in the following table. To find the factorial of a number, multiply the number with the factorial value of the previous number. For example, to know the value of 6! multiply 120 (the factorial of 5) by 6, and get 720. For 7! multiply 720 (the factorial value of 6) by 7, to get 5040.

n | n! | ||

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 |

**Summary**

**The factorial (denoted or represented as n!) for a positive number or integer (which is denoted by n) is the product of all the positive numbers preceding or equivalent to n (the positive integer).****In mathematics, there are a number of sequences that are comparable to the factorial. They include Double Factorials, Multi-factorials, Primorials, Super-factorials, and Hyper-factorials.****The factorial of 0 is equal to 1 (one).**

**Formula for n Factorial**

The formula for n factorial is:n!=n×(n−1)!n!=n×(n−1)!

n!=n×(n−1)!

This means that the factorial of any number is, the given number, multiplied by the factorial of the previous number. So, 8!=8×7!…… And 9!=9×8!…… The factorial of 10 will be 10!=10×9!…… Like this if we have (n+1) factorial then it can be written as, **(n+1)!=(n+1)×n!**

**Factorial of Negative Numbers**

**Use of Factorial**

One area where factorials are commonly used is in permutations & combinations.Now,** permutation** is an ordered arrangement of outcomes and it can be calculated with the formula:

n | n! |

1 | 1 |

2 | 2 |

3 | 6 |

4 | 24 |

5 | 120 |

6 | 720 |

7 | 5040 |

8 | 40,320 |

9 | 362,880 |

10 | 3,628,800 |

11 | 39,916,800 |

12 | 479,001,600 |

13 | 6,227,020,800 |

14 | 8,717,8291,200 |

15 | 1,307,674,368,000 |

## Solved Examples on Factorial

**Example 3: How many 5-digit numbers can be formed using the digits 1, 2, 5, 7, and 8 in each of which no digit is repeated?Solution:**The given 5 digits (1, 2, 5, 7 and 8) should be arranged among themselves in order to get 5-digit numbers.The number of ways for doing this can be done using the factorial.5! = 5 × 4 × 3 × 2 × 1 = 120

Therefore, the required number of 5-digit numbers is 120.

**Example 4: In how many ways can eight people line up from left to right for a group photo?Solution:** Number of ways 8 people line up

= 8!

= 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

= 40,320

Therefore, Eight people can line up in 40,320 ways.

## FAQs on Factorial

### What is the Relation Between Factorial, Permutation, and Combination?

Factorial is a function which is used to find the number of possible ways in which a selected number of objects can be arranged among themselves. This concept of factorial is used for finding permutations and combinations of numbers and events.

### What is n+1 Factorial?

n+1 factorial can be calculated as (n+1)! = (n+1)n!

### What is the Factorial of 10?

10! can be calculated as 10!=10×9!=10×362,880=3,628,800.

### What is the Factorial of a Number?

In mathematics, factorial of a number means multiplication of a positive integer with every integer less than that. So, n!= n × (n-1) × (n-2) × (n-3) × ….. × 3 × 2 × 1.

### What are Factorials Used for?

Factorials are used to find the number of patterns, solving permutation and combination problems, finding out probability of events, etc.

### What is the Factorial Symbol?

The symbol used to represent factorial is ‘ ! ‘.

### What is Factorial Notation?

Factorial notation is the expanded form of the factorial of a number. So, 3! is 3 × 2 × 1, 5! is 5×4×3×2×1, and so on.

### Defining the Factorial

The function of a factorial is defined by the product of all the positive integers before and/or equal to *n,* that is:

*n!* = 1 ∙ 2 ∙ 3 ∙∙∙ (*n*-2) ∙ (*n* -1) ∙* n*,

when looking at values or integers greater than or equal to 1. It can then be written as:

The equation above is written according to the pi product notation and results in the recurring relation seen below:

*n! = n *∙* (n –* 1) *!.*

Some examples of the notation can be seen below:

- 4! = 4 ∙ 3!
- 7! = 7 ∙ 6!
- 80! = 80 ∙ 79!, etc.

### Factorial Table

The table below gives an overview of the factorials for integers between 0 and 10:

### Factorial of 0 (Zero)

It is widely known that the factorial of 0 is equal to 1 (one). It can be denoted as:

**0! = 1**

There are several reasons to justify the notation and definition stipulated above. Firstly, the definition provides an allowance for a compact expression of a considerable number of formulae, including the exponential function, and the definition creates an extension of the recurrence relation to 0.

In addition, where n = 0, the definition of its factorial (n!) encompasses the product of no numbers, meaning that it is equivalent to the multiplicative identity in broader terms.

Moreover, the definition of the zero factorial includes only one permutation of zero or no objects. Lastly, the definition also validates a number of identities in combinatorics.

### Definitions to Note in Relation to the Zero Factorial

**Combinatorics**: An area in mathematics that focuses on counting.**Permutation**: In mathematics, permutation refers to the arrangement of the members of a set into a linear order or sequence.**Recurrence relation**: Recurrence relation, in mathematics, refers to an equation that defines a sequence or vast array of values recursively. Recursion means defining something in terms of itself.

### Other Sequences Similar to the Factorial

In mathematics, there are a number of sequences that are comparable to the factorial. They include:

**Double Factorials**, which are used to simplify trigonometric integrals.**Multi-factorials**, which can be denoted with multiple exclamation points.**Primorials**, which entail getting the product of the prime numbers, which are less than or equal to*n*.**Super-factorials**, which are defined as the product of the first*n*factorials.**Hyper-factorials**, which are a result of multiplying a number of consecutive values ranging from 1 to*n*.

## A Small List

n | n! |
---|---|

0 | 1 |

1 | 1 |

2 | 2 |

3 | 6 |

4 | 24 |

5 | 120 |

6 | 720 |

7 | 5,040 |

8 | 40,320 |

9 | 362,880 |

10 | 3,628,800 |

11 | 39,916,800 |

12 | 479,001,600 |

13 | 6,227,020,800 |

14 | 87,178,291,200 |

15 | 1,307,674,368,000 |

16 | 20,922,789,888,000 |

17 | 355,687,428,096,000 |

18 | 6,402,373,705,728,000 |

19 | 121,645,100,408,832,000 |

20 | 2,432,902,008,176,640,000 |

21 | 51,090,942,171,709,440,000 |

22 | 1,124,000,727,777,607,680,000 |

23 | 25,852,016,738,884,976,640,000 |

24 | 620,448,401,733,239,439,360,000 |

25 | 15,511,210,043,330,985,984,000,000 |

As you can see, it gets big quickly.

If you need more, try the Full Precision Calculator.

**Some Applications of Factorial Value**

Some applications of factorial in mathematics are as follows:

**1)Recursion**

In the recursive definition of a number, we may use factorial. A number can be expressed in an expression containing the number only.

p!=p×(p–1)×(p–2)×(p–3)..(p−(p–2))×(p–(p–1))

**2) Permutations**

Arrangement of given r things out of total n things when order is strictly important.

**4) Probability Distributions**

There are various probability distributions like binomial distribution which include the use of factorial. To find the probability of an event, the concept of permutations and combinations is used a lot.

**5) Number Theory**

Factorials value are used extensively in number theory and also for approximations.

**Solved Examples for Factorial Formula**

Q.1: What is the value of 8!?

Solution: The formula for factorial is,

p!=p×(p−1)×(p−2)×(p−3)…×2×1

8!=8×(8−1)×(8−2)…×2×1

=8×7×6×….3×2×1

= 40320

^{9!}/_{5!}=3024

## C program for factorial calculation

double factorial(unsigned int n)

{

double fact=1.0;

if( n > 1 )

for(unsigned int k=2; k<=n; k++)

fact = fact*k;

return fact;

}

## Formula for Factorial n

The **Factorial Formula** is given as,

n Factorial Formula | n! = 1 × 2 × 3 × ….. × (n − 1) × n |

### Solved Examples Using Factorial Formula

**Question 1: **What is 8!?

**Solution:**

The formula formula for factorial is,

n! = 1 × 2 × 3 × ………. × (n-1) × n

8! = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8

8! = 40320

### Factorial Examples

**Example 1:**

What is the factorial of 6?

**Solution:**

We know that the factorial formula is

**n! = n × (n – 1) × (n – 2) × (n – 3) × ….× 3 × 2 × 1**

So the factorial of 6 is

6! = 6 × (6 -1) × (6 – 2) × (6 – 3) × (6 – 4) × 1

6! = 6 × 5 × 4 × 3 × 2 ×1

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!?

## Frequently Asked Questions on Factorial

### What is a factorial of 10?

The value of factorial of 10 is 3628800, i.e. 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800.

### What is the meaning of 5 factorial?

The meaning of 5 factorial is that we need to multiply the numbers from 1 to 5. That means, 5! = 5 × 4 × 3 × 2 × 1 = 120.

### What is the symbol of factorial?

The factorial function is a mathematical formula represented by an exclamation mark “!”. For example, the factorial of 8 can be represented as 8! and it is read as eight factorial.

### What is a factorial of 0?

The value of factorial of 0 is 1, i.e. 0! = 1.

### What is the value of 7!?

The value of 7! is 5040, i.e. 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040.

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