1119 LearnersLast updated on June 5th, 2025
Factorial
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 utilized to determine the possible ways of arranging things. In this topic, you will easily learn about factorials and their applications in detail.
What is Factorial?
Factorials are fundamental concepts in learning complex topics like algebra and probability. A factorial is a function that multiplies a number by every whole number lower than the number, until 1. We use the symbol “!” to denote the factorial.
For example, the factorial for 5 is:
5! = 5 × 4 × 3 × 2 × 1 = 120
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Factorial Table
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)
For example:
- To find the factorial of 6, multiply 6 by the factorial of 5:
6! = 6 × 120 = 720 (5! = 120)
- Similarly, the factorial of 7:
7! = 7 × 720 = 5040 (6! = 720)
| 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 | =3×3! | =6 |
| 4 Factorial | 4×3×2×1 | =4×3! | =24 |
| 5 Factorial | 5×4×3×2×1 | =5×4! | =120 |
Factorial of 0 (Zero)
It is easy to assume that 0 is the factorial of 0. But this assumption is wrong. 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
Factorial of n
The factorial of a number n is the product of the first n natural integers can be expressed as:
n! = n × (n –1) × (n – 2) ×… × 3 × 2 × 1.
The n factorial can be mathematically represented as the product of the given number by the factorial of the preceding number:
n! = n × (n – 1)!
Factorial of Negative Numbers
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 = (division by 0 is impossible)
Thus, factorials are undefined for negative numbers.
How to Calculate Factorial of Numbers
As we have learned, the factorials of n are represented as n! And is determined using the formula n! = n × (n – 1)!
For example: If 7! = 5,040, find the value of 8!
We solve it as:
40,320 = 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 |
Real-Life Applications of Factorial
Factorials are of immense significance in various real-life situations. Let’s now look at a few such examples:
- Factorials play a vital role in determining the total number of ways things can be arranged, such as seating arrangements and the arrangement of books on a shelf.
- They can be applied to calculate the total possible outcomes in events influenced by probability, such as outcomes in lotteries.
- In scientific scenarios, they are used for gene arrangements or in DNA sequencing.
- In sports, they are used in planning and arranging tournaments.
- They are utilized in detecting the risk factors in finance by determining the possible outcomes in stock markets.
Common Mistakes and How to Avoid Them in Factorials
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:
Mistake 1
Incorrectly Interpreting Factorial Notation
Some students might misunderstand the factorial notation and assume that any number n! represents multiplying the number by itself multiple times.
Example: 4! = 4 × 4 × 4 × 4 (incorrect).
Understand that the factorial of any number is the product of the number and all whole numbers less than or equal to it, until 1. For example, 4! = 4 × 3 × 2 × 1 = 24
Mistake 2
Using Factorials for Negative Numbers
Attempting to determine factorials for negative numbers leads to confusion.
Keep in mind that factorials can only be applied to non-negative numbers. For negative values, factorials are undefined.
For example: (– 2)! = undefined (since division by zero is impossible).
Mistake 3
Wrong Division of Factorials
One common mistake is that assuming n!/n = (n – 1)! applies to every number.
For example: 0!/0 = (0 – 1)! is incorrect because division by 0 is undefined.
Remember that the division of factorials works only when the number is positive.
For example: 6!/6 = (6 × 5 × 4 × 3 × 2 × 1)/6 = 120/6 = 20
Mistake 4
Misinterpretation About 0!
Students often misunderstand that 0! is zero, considering that the product of zero and any other number results in zero.
Understand the mathematical derivation of 0! = 1 by definition.
Mistake 5
Forgetting Multiplication Steps
Some students might forget the steps in calculation and cause errors in finding the solution.
To avoid miscalculation, always ensure that the calculations are performed step-by-step.
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Solved examples of Factorial
Problem 1
Determine the value of (5! ÷ 4! × 3!).
(5! ÷ 4! × 3!) = 30
Explanation
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
Problem 2
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.
Explanation
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.
Problem 3
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.
Explanation
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.
Problem 4
Determine the value of 4!10!
4!10! = 87,091,200
Explanation
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
Problem 5
Find the value of (12! – 8!)
(12! – 8!) = 478,961,280
Explanation
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
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FAQs on Factorials
1.What do you mean by a factorial?
2.Is the value 0! Defined?
3.Give one real-life example of factorials.
4.Can we calculate the factorials for negative numbers?
5.How can children in United Kingdom use numbers in everyday life to understand Factorial?
6.What are some fun ways kids in United Kingdom can practice Factorial with numbers?
7.What role do numbers and Factorial play in helping children in United Kingdom develop problem-solving skills?
8.How can families in United Kingdom create number-rich environments to improve Factorial skills?
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