Factorial
2026-02-28 13:01 Diff
https://brightchamps.com/en-us/math/numbers/factorial

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Last updated on December 2, 2025

Imagine you are taking a number and multiplying it by every whole number before it until you reach 1; that result is the factorial. Factorials help you discover how many ways things can be arranged. In this article, we will explore the concept in detail.

What is a Factorial?

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Factorial is the way of multiplying a number by every whole number below it, all the way down to 1. We show that a factorial is denoted by the symbol ‘!’, which is the one above the number 1 on a computer keyboard. When we say n! (read as “n factorial”), which means we multiply all the positive integers from 1 up to n.

So, the factorial definition can be written as:

\(\text{n! = 1 × 2 × 3 × ... × n or, }\\[1em] \text{n! = n × (n – 1) × (n – 2) × ... × 3 × 2 × 1}\)

Example:

Find the value of 5! (5 factorial).

Answer

\(5! = 5 × 4 × 3 × 2 × 1\)

5! = 120

So, the value of 5! is 120.

Formula for Factorial

The factorial of a number n is found by multiplying all the whole numbers from n down to 1. The formula is:

For any whole number n ≥ 1, the factorial can also be written using the product notation:

From these formulas, we can get a useful pattern called the recurrence relation, which shows how each factorial is built using the previous one:

n! = n × (n − 1)!

This means you can find the larger factorial by multiplying the number n by the factorial of the number just before it.

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., \(n! = n \times (n-1) \times (n-2) \times \dots \times 1 \)

For example, to find the factorial of 6, multiply 6 by the factorial of 5:

\(6! = 6 \times 120 = 720 \quad (5! = 120) \)

Similarly, the factorial of 7:

\(7! = 7 \times 720 = 5040 \quad (6! = 720) \)

n Factorial \(\mathbf{n (n-1) (n-2) \dots 1 }\) \(\mathbf{n! = n \times (n-1)! }\) 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

\(5 × 4 × 3 × 2 × 1\) \(= 5 × 4!\) \(= 120\)

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What is the Subfactorial of a Number?

A subfactorial of a number is written as !n. It tells us how many ways we can rearrange the n objects so that none of them stay in their original position. In other words, it counts the number of derangements, shuffled arrangements where every item is moves to a different place.

The formula for finding the subfactorial of n is:
 

\(!n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}\)

Factorial of 0 (Zero)

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:

\(3! = \frac{4!}{4} \)

\(2! = \frac{3!}{3} \)

\(1! = \frac{2!}{2} \)

\(0! = \frac{1!}{1} \)

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.

\(3! = 3 \times 2 \times 1 = 6 \)

\(2! = \frac{3!}{3} = \frac{6}{3} = 2 \)

\(1! = \frac{2!}{2} = \frac{2}{2} = 1 \)

\(0! = \frac{1!}{1} = \frac{1}{1} = 1 \)

\((–1)! = 0! ÷ 0 = 1 ÷ 0\) (undefined, since division by zero 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 \(n! = n × (n – 1)!\)
 

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

Tips and Tricks to Master Factorial

For developing a deeper understanding of factorials, follow these tips and tricks given below: 

  • Start by recognizing that a factorial is the product of all positive integers up to a given number. 
     
  • Remember the recursive relationship  \(n! = n × (n - 1)!\). This means you can compute larger factorials by building upon smaller ones. 
     
  • When dividing factorials, cancel common terms to simplify calculations. 
     
  • Many calculators have a factorial function (\(n!\)). Utilize this feature to quickly compute factorials without manual multiplication. 
     
  • Factorials grow rapidly and from them try to recognize the patterns. For example, \(10! = 3,628,800\). 
     
  • Explain that the factorial notation (n!) means multiplying all positive integers from 'n' to '1'.
     
  • Start with 1!, 2!, 3 Factorial (3!), 4!. Small numbers build confidence before the larger ones.
     
  • Use the fundamental factorial rule: n! = n × (n − 1)!. This shows how each factorial uses the previous one.
     
  • Most calculators (and online tools) have a factorial calculator function. Let children try the manual steps first, then verify using the tool.

Common Mistakes and How to Avoid Them in Factorials

The concept of factorial is an important 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:

Real-World 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 used in finance to evaluate risk factors by analyzing possible outcomes in stock markets.

Problem 1

Determine the value of (5! ÷ 4! × 3!).

Okay, lets begin

\((5! ÷ 4! × 3!) = 30\)

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:


\((5! ÷ 4! × 3!) = (120 ÷ 24) × 6\)


\(= 5 × 6\)


\(= 30\)

Well explained 👍

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?

Okay, lets begin

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.

Well explained 👍

Problem 3

How many different ways can the letters in the word "EDUCATION" be arranged if all letters are used?

Okay, lets begin

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.

Well explained 👍

Problem 4

Determine the value of 4!10!

Okay, lets begin

\(4! × 10! = 87,091,200\)

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:


\( 4!10! = 24 × 3,628,800 = 87,091,200\)

Well explained 👍

Problem 5

Find the value of (12! – 8!)

Okay, lets begin

\(12! – 8! = 479,001,600 – 40,320 = 478,961,280.\)

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:


\(12! – 8! = 479,001,600 – 40,320\)


\(= 478,961,280\)

Well explained 👍

FAQs on Factorials

1.What do you mean by a factorial?

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: 
\(5! = 5 × 4 × 3 × 2 × 1 = 120.\)

2.Is the value 0! Defined?

Yes, the value of \(0!\) is defined. \( 0!= 1\).

3.Give one real-life example of factorials.

Factorials are used in determining the total number of arrangements of things like books on a shelf.

4.Can we calculate the factorials for negative numbers?

No, factorials cannot be calculated as they are undefined for negative numbers.

Hiralee Lalitkumar Makwana

About the Author

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.

Fun Fact

: She loves to read number jokes and games.