Factorial

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.
 

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.

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:

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

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:
 

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

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:

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

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

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

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

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:

\(n! = n × (n – 1)!\)

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. 

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:

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.

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.