Factorial

The Factorial of a whole number 'n' is defined as the product of that number with every whole number less than or equal to 'n' 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!. The study of factorials is at the root of several topics in mathematics, such as number theory, algebra, geometry, probability, statistics, graph theory, discrete mathematics, etc.

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. In the year 1808, when a mathematician from France, Christian Kramp, came up with the symbol for factorial: n!. Thinking about how to calculate the factorial of a number? Let's learn.

1.	What is Factorial?
2.	n Factorial Formula
3.	0 Factorial
4.	Factorial of Hundred
5.	Factorial of Negative Numbers
6.	Use of Factorial
7.	How to Calculate Factorial?
8.	FAQs on Factorial

The factorial of a whole number is the function that multiplies the number by every natural number below it. Symbolically, a factorial can be represented by using the symbol "!". This symbol lies on the same key above "1" on a computer keyboard. "n factorial" is the product of the first n natural numbers and is represented as n!

Factorial Meaning

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)

Example: 5 factorial, that is, 5! can be written as: 5! = 5 × 4 × 3 × 2 × 1 = 120.

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, get 5040. i.e., n! = n × (n - 1)!

n Factorial	n (n - 1) (n - 2) ....1	n! = n × (n - 1)!	Result
1 Factorial	1	1	1
2 Factorial	2 × 1	= 2 × 1!	= 2
3 Factorial	3 × 2 × 1	= 3 × 2!	= 6
4 Factorial	4 × 3 × 2 × 1	= 4 × 3!	= 24
5 Factorial	5 × 4 × 3 × 2 × 1	= 5 × 4!	= 120

The formulas for n factorial are:

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

The first formula directly follows from factorial definition whereas the second formula 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!. Let us see some examples.

5 Factorial

The value of 5 factorial is 5×4×3×2×1 which is equal to 120. We can evaluate it using the factorial formula as well. 5! = 5 × 4! = 5 × 24 = 120.

10 Factorial

10 factorial is nothing but 10 × 9 × 8 × 7 × 6 × 5 × 4 ×3 × 2 × 1 = 3,628,800.

Zero factorial is interesting, and its value is equal to 1, i.e., 0! = 1. Yes, the value of 0 factorial is NOT 0, but it is 1.

Let us see how this works:

1! = 1
2! = 2 × 1 = 2
3! = 3 × 2 × 1 = 3 × 2! = 6
4! = 4 × 3 × 2 × 1 = 4 × 3! = 24

Let’s go to the basic formula of factorial n! = n × (n - 1)! How to find 3! What you do is 4! / 4. Similarly, 2! is 3! / 3, and so on. Now, let’s look at the pattern:

In this way, we could prove that 0 factorial is 1.

Alternative Way of Proving 0! = 1

In permutations, we would study that n! is the number of ways of arranging 'n' different things among themselves. If we look factorial in this way, 1! = 1 as there is only 1 arrangement possible with 1 thing. In the same way, 0! = 1.

100 factorial = 100 × 99 × 98 × .... × 3 × 2 × 1 = 9.332621544 E+157. This product is too big to calculate manually and hence a calculator is used. Here are some facts about hundred factorial:

• 100 factorial has 24 trailing zeros in it.
• The total number of digits in 100! is 158.
• The exact value of 100 factorial is 93, 326, 215, 443, 944, 152, 681, 699, 238, 856, 266, 700, 490, 715, 968, 264, 381, 621, 468, 592, 963, 895, 217, 599, 993, 229, 915, 608, 941, 463, 976, 156, 518, 286, 253, 697, 920, 827, 223, 758, 251, 185, 210, 916, 864, 000, 000, 000, 000, 000, 000, 000, 000 (158 digits in total).

Can we have factorials for numbers like −1, −2, etc? Let's start with 3! = 3 × 2 × 1 = 6

3! = 3 × 2 × 1 = 6
2! = 3! / 3 = 6 / 3 = 2
1! = 2! / 2 = 2 / 2 = 1
0! = 1! / 1 = 1 / 1 = 1
(- 1)! = 0! / 0 = 1 / 0 = dividing by zero is undefined

And from here on down all integer factorials are undefined. So, negative integer factorials are undefined.

One area where factorials are widely used is in permutations & combinations.

• Permutation is an ordered arrangement of outcomes and it can be calculated with the formula: ⁿ P_r = n! / (n - r)!
• Combination is a grouping of outcomes in which order does not matter. It can be calculated with the formula: ⁿC_r = n! / [ (n - r)! r!]

In both of these formulas, 'n' is the total number of things available and 'r' is the number of things that have to be chosen. Let us understand this by the following examples.

Example 1: In a group of 10 people, $200, $100, and $50 prizes are to be given. In how many ways can the prizes be distributed?

Solution:

This is permutation because here the order of distribution of prizes matters. It can be calculated as ¹⁰P₃ ways.

¹⁰P₃ = (10!) / (10 - 3)! = 10! / 7! = (10 × 9 × 8 × 7!) / 7! = 10 × 9 × 8 = 720 ways.

Example 2: Three $50 prizes are to be distributed to a group of 10 people. In how many ways can the prizes be distributed?

Solution:

This is a combination because here the order of distribution of prizes does not matter (because all prizes are of the same worth). It can be calculated using ¹⁰C₃.

¹⁰C₃ = (10!) / [ 3! (10 - 3)!] = 10! / (3! 7!) = (10 × 9 × 8 × 7!) / [(3 × 2 × 1) 7!] = 120 ways.

The factorial of n is denoted by n! and calculated by multiplying the integer numbers from 1 to n. The formula for n factorial is n! = n × (n - 1)!.

Example: If 8! is 40,320 then what is 9!?

Solution:

9! = 9 × 8! = 9 × 40,320 = 362,880

Now, let us look at a factorial table given below that shows the values of factorial for the first 15 natural numbers:

n Factorial	Value
1 Factorial	1
2 Factorial	2
3 Factorial	6
4 Factorial	24
5 Factorial	120
6 Factorial	720
7 Factorial	5040
8 Factorial	40,320
9 Factorial	362,880
10 Factorial	3,628,800
11 Factorial	39,916,800
12 Factorial	479,001,600
13 Factorial	6,227,020,800
14 Factorial	8,717,8291,200
15 Factorial	1,307,674,368,000

☛Related Topics:

• Factorial Calculator
• What is the Factorial of 9?
• What is the Factorial of 20?

Important Notes on Factorial:

• The factorial of any whole number can be calculated using n! = n × (n - 1)!.
• The value of zero factorial is one, i.e., 0! = 1.
• Negative integer factorials are undefined.
• Permutation & Combination can be calculated using factorials: ⁿP_r = n! / (n - r)! & ⁿC_r = n! / [(n - r)! r!].

 

Where do We Use Factorials?

Factorial is a function that 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 Factorial Notation?

Factorial notation is writing the product of consecutive whole numbers in the form of a factorial. So, 3 × 2 × 1 = 3! (3 factorial), 6 × 5 × 4 × 3 × 2 × 1 = 6! (6 factorial), and so on.

What is n+1 Factorial?

n+1 factorial can be calculated using (n+1)! = (n+1)n!. This is because the factorial of a number is the number multiplied by its previous number's factorial.

What is the Factorial Symbol?

The symbol used to represent factorial is ' ! '. For example "9 factorial" is written as 9!.

What is the Factorial of Hundred?

The factorial of 100 is written as 100! and its value is 100 · 99 · 98 · ... · 2 · 1 = 9.332621544 E+157. It has 158 digits in it with 24 trailing zeros.

What is the Factorial of 10?

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

How to Find Factorial of a Number?

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

What are the Applications of Factorials?

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