Summarize this article:
Last updated on September 11, 2025
A factorial is what we get when we multiply a number by every whole number that comes before it, all the way down to 1. Factorials are used to determine the number of possible arrangements of objects. In this topic, you will easily learn about factorials and their applications in detail.
The factorial table showcases the numbers and their factorial values. As seen below, we determine the factorial of a given number by multiplying it by the factorial of the preceding number.
i.e., n! = n × (n – 1) × (n – 2) × … × 1
For example:
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 |
It is easy to assume that the factorial of 0 is 0, but this is incorrect. The factorial of 0 is 1 which can be written as:
0! = 1.
Factorials often follow a pattern:
1! = 1
2! = 2 × 1 = 2
3! = 3 × 2 × 1 = 3 × 2! = 6
4! = 4 × 3 × 2 × 1 = 4 × 3! = 24
To understand the zero factorial better, let’s look at the following method:
We find 3! by dividing the factorial of the succeeding number by that number:
3! = 4!/ 4
2! = 3!/ 3
1! = 2!/ 2
0! = 1!/ 1
There is a common misconception that factorials include negative numbers. We will now learn why factorials are undefined for negative numbers. Here, we start with the factorial of 3.
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 (undefined, since division by zero is impossible).
Thus, factorials are undefined for negative numbers.
As we have learned, the factorials of n are represented as n! And is determined using the formula n! = n × (n – 1)!
If 7! = 5,040, then 8! = 8 × 7! = 8 × 5,040 = 40,320.
The table below shows the factorials of the first 15 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 | 40320 |
9 Factorial | 362880 |
10 Factorial | 3628800 |
11 Factorial | 39916800 |
12 Factorial | 479001600 |
13 Factorial | 6227020800 |
14 Factorial | 87178291200 |
15 Factorial | 1307674368000 |
Factorials are of immense significance in various real-life situations. Let’s now look at a few such examples:
The concept of factorial is important in number theory and has several applications. However, students might make mistakes when solving problems related to it. Here are a few common mistakes and the easy ways to avoid them:
Determine the value of (5! ÷ 4! × 3!).
(5! ÷ 4! × 3!) = 30
Let’s first calculate the factorials of 5, 4, and 3 separately:
5! = 5 × 4 × 3 × 2 × 1 = 120
4! = 4 × 3 × 2 × 1 = 24
3! = 3 × 2 × 1= 6
We now substitute these values:
(5! ÷ 4! × 3!) = (120 ÷ 24) × 6
= 5 × 6
= 30
Alex has 8 different books and wants to organize them on a shelf. In how many possible ways he can organize these books?
Alex can organize the books in 40,320 different ways.
To find the possible ways in which Alex can organize the books, we use the formula:
n! = n (n –1) (n – 2)... 3 × 2 × 1
Given that, there are 8 books:
So the total number of arrangements is:
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 40,320
Therefore, Alex can organize the books in 40,320 different ways.
How many different ways can the letters in the word "EDUCATION" be arranged if all letters are used?
The word“ EDUCATION” can be arranged in 362,880 different ways.
All the letters in the given word are unique, so the total number of ways to arrange them is:
n! = n (n –1) (n – 2)... 3 × 2 × 1
9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880
So, there are 362,880 different arrangements.
Determine the value of 4!10!
4! × 10! = 87,091,200
Let’s first calculate the factorials of 4 and 10 separately:
4! = 4 × 3 × 2 × 1 = 24
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
We’ll now multiply the factorials:
4!10! = 24 × 3,628,800 = 87,091,200
Find the value of (12! – 8!)
12! – 8! = 479,001,600 – 40,320 = 478,961,280.
We will first find the factorials of 12 and 8 separately:
12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 479,001,600
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320
Now, substitute these values:
12! – 8! = 479,001,600 – 40,320
= 478,961,280
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.