Factorial
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!. 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.
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.  Calculation of Factorial 
8.  FAQs on Factorial 
What is 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 "!". 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(n1)(n2)…………………….(3)(2)(1)
For example, 4 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  n! (n factorial) 
n! = n × (n  1)!  Result 

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 
n Factorial Formula
The formula for n factorial is: 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!. 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.
0 Factorial
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 its 1.
Let us see that 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.
Factorial of Hundred
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).
Factorial of Negative Numbers
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.
Use of Factorial
One area where factorials are commonly 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: ^{n}C_{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 ^{10}P_{3} ways.
^{10}P_{3} = (10!) / (10  3)! = 10! / 7! = (10 × 9 × 8 × 7!) / 7! = 10 × 9 × 8 = 720 ways.
Example 2: Three $50 prizes are to be distributed in 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 ^{10}C_{3}.
^{10}C_{3} = (10!) / [ 3! (10  3)!] = 10! / (3! 7!) = (10 × 9 × 8 × 7!) / [(3 × 2 × 1) 7!] = 120 ways.
Calculation of Factorial
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  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 
☛Related Topics:
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: ^{n}P_{r} = n! / (n  r)! &^{ n}C_{r} =n! / [(n  r)! r!].
Factorial Examples

Example 1: Evaluate the expression involving factorials: 10!/(4! × 6!).
Solution:
10!/(4! × 6!) = (10 × 9 × 8 × 7 × 6!) / (4! × 6!)
= (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1)
= 210Answer: Therefore, the value of expression 10!/(4! × 6!) is 210.

Example 2: Find the value of 5! (6  3)!.
Solution:
5! (6−3)! = 5! × 3!
Now, we will calculate these factorials.
= (5 × 4 × 3 × 2 × 1)(3 × 2 × 1)
= 120 × 6
= 720
Answer: Therefore, the value of 5! (6  3)! is 720.

Example 3: How many 5digit 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 all possible 5digit numbers.
The number of ways for doing this can be done using the factorial.
5! = 5 × 4 × 3 × 2 × 1 = 120
Answer: Therefore, the required number of 5digit 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 is the number of ways in which they can be arranged among themselves and this is nothing but 8 factorial.
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320
Answer: Therefore, eight people can line up in 40,320 ways.
Practice Questions on Factorial
FAQs on Factorial
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 the Meaning of Factorial of a Number?
Factorial in math is one of the operations (denoted by the symbol "!") and the factorial of a number is the product of the number with all positive integers less than that number. For example 8! (which is read as 8 factorial) = 8 · 7 · 6 ·5 · 4 · 3 · 2 · 1 = 40,320.
What is n+1 Factorial?
n+1 factorial can be calculated as (n+1)! = (n+1)n!.
What is the Factorial Symbol?
The symbol used to represent factorial is ' ! '. For example "9 factorial" is written as 9!.
What is Factorial of 100?
The factorial of 100 is written as 100! and its value is 100 · 99 · 98 · ... · 2 · 1 = 9.332621544 E+157.
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, the factorial of a number means the multiplication of the number with every positive integer less than that. So, n!= n × (n1) × (n2) × (n3) × ..... × 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.
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.
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