Factorial of a Number: Definition, Formula & Examples

Factorial is an essential function applied to find how many things can be arranged or the ordered set of numbers. Daniel Bernoulli discovered the well known interpolating function of the factorial function. The factorial of a number is one of those concepts which are used in several other mathematical concepts, including probability, permutations and combinations, and more.

It’s not that finding out factorial is a complex task, rather it is one of the simplistic mathematical approaches to find the number of ordered arrangements. In this article, we are going to emphasize everything about factorial and help you understand what is the utility and specifications of factorial value. Without further ado, let’s start and grab everything we can about the factorial.

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Brief Introduction of Factorial

A factorial of a whole number ‘n’ is defined as the product of every whole number below the number including it. The factorial is generally denoted by (!). 

For example, if it asked to find 6 factorials, then 6!=6×5×4×3×2×1. This means that 6! Is equal to the product of numbers starting from 6 till it reaches one, resulting in a product of 720.

The factorial value is defined by Fabian Stedman, a British author as synonymous with change ringing. But it was in 1808 that the symbol of factorial, the exclamation mark (!) was given by Christian Kramp. You’ll be amused to know factorial properties encompasses the foundations for several other mathematical topics such as statistics, probability, geometry, numbers, etc.

How to Find the Factorial of a Number?

As discussed above, factorial is simply the product of all natural numbers below ‘n’ till it reaches one. One thing that needs to be kept in mind is that the product is inclusive of n.

So, if you’re seeking any factorial formula, then the required formula can be stated as follows:

Factorial of a given number, n= n!= n×(n-1)×(n-2)×(n-3)×..............................×1

Or, we can say that,

n!= n× (n-1)!

This simply denotes that the factorial of ‘n’ number is equivalent to the product of that number and the factorial of a number below it. For instance, 8!= 8×7! And 11!= 11×10! And so on. Therefore, if you’re asked to find the factorial of (n+2), then it is equal to (n+2)×(n+1)!

What is the Factorial of 0?

Till now, we understood what is the factorial and how to find the factorial of a number. You simply have to do a product of a series of ‘n’ numbers till it comes down to one.

However, what if you’re asked to find a factorial of zero? Then how can you find it? Zero is an interesting and little tricky concept.

Whenever you’re asked to find the factorial of zero, then it is equal to one; 0!=1

Why? Let’s recall the factorial formula once again and understand why such is the case with 0! 

1! = 1

2! = 2×1=2×1!=2

3! = 3×2×1=3×2!=6

4! = 4×3×2×1=4×3!=24

As per this, we understood to find the factorial of a given number, the multiplication of a given number with the factorial of the number below it answers.

This basic formula leads us to know that to find 4! you can do, 5! ÷ 5 which results ultimately to 4!

Following this, you will find 0!= 1! ÷ 1= 1.

How to Find a Factorial of a Negative Number?

Till now we’ve learned what is factorial of a positive number. However, have you wondered what if you are asked to find a factorial of a negative number? Is it possible? Let’s find out.

3! = 3×2! = \[\frac{4!}{4}\] = \[\frac{24}{4}\] = 6

2! = 2×1! = \[\frac{3!}{3}\] = \[\frac{6}{3}\] =2

1! = 1×0! = \[\frac{2!}{2}\] = \[\frac{2}{2}\] =1

0! = \[\frac{1!}{1}\] =  \[\frac{1}{1}\] = 1

-1! = \[\frac{0!}{0}\] = \[\frac{1}{0}\] = N.D.

From this, we can conclude that the factorial of negative numbers is not defined.

What are the Uses of Factorial?

• Permutation: It tells us the number of ways in which the outcomes can be arranged. The formula of permutation encompasses the use of the factorial formula. It can be calculated as:

nPr = \[\frac{n!}{\left ( n-r \right )!}\]

• Combination: It tells us in how many ways we can obtain a certain combination of outcomes. Here, the order isn't a lot of concern. It can be calculated as follows:

nCr = \[\frac{n!}{r!(n-r)!}\]

Solved Example

Question: Evaluate the expression: \[\frac{7!}{4!\times 2!}\]

1. 110

2. 105

3. 130

4. 125

Answer: (b)

By using the formula of a factorial, (n-1)! = (n-1) × n!

\[\frac{7!}{4!\times 2!}\] = \[\frac{7\times 6\times 5\times 4!}{4!\times 2\times 1}\] = \[\frac{7\times 6\times 5}{2}\] = \[7\times 3\times 5\] = 105

Fun Facts About Factorial

• Very few people know that the number counts in the factorial of 22 and 23 are 22 and 23 respectively.

• If you want to calculate the number of zeros at the end of the factorial of any whole number, then you can do so by using an easy formula? That is \[\frac{n}{4}\] . Interesting isn't it.

Conclusion

• Factorial of a number is the multiplicative series of a number n with the factorial of the (n-1). 

• In the case of zero factorial, the value is 1. 

• You can find out the factorial of any positive whole number. 

• However, negative numbers don't have any factorial value. They are not defined.