Factorial: Meaning, Formula, Values for Numbers 1 to 10

In this article we will learn all about what is factorial in mathematics? Factorial definition is very simple - Factorial is an important function in mathematics that is related to any positive integer. The factorial of a number is the product of the given number multiplied by all natural numbers less than the given number up to the number 1. The factorial plays an important role in statistical operations, especially for permutations and combinations.

Factorial Numbers in Real Life

Did you know that factorial numbers are also real for fact? When arranging a deck of cards or solving a puzzle, the game uses the properties of the factorial in calculating how many ways are possible to use. This helps game makers create how many possibilities a player has of winning. Pretty awesome, right?

What is a Factorial Formula? 

The factorial of a given number is calculated by multiplying all the natural numbers less than or equal to that number and the last number should be 1. The factorial of a number is expressed by the symbol ‘!’ placed after the number. For example, the factorial of 5 is expressed by the symbol 5! Let’s explore the factorial formula.

The formula of factorial number ‘n’ is expressed below.

n! = n x (n-1) x (n-2) x (n-3) …….. 3 x 2 x 1

Therefore, the value of factorial 5 will be: 5! = 5 x 4 x 3 x 2 x 1 = 120

Learn the Factorial Formula

Let us consider that you have 4 cupcakes. You would like to know how many possible different arrangements can be according to the line. The definition of formula in factorial: 

4! = 4 × 3 × 2 × 1 = 24. 

So placing them yields 24 different arrangements! One uses it to solve many fun problems with a knowledge of the formula of factorial. It is like a magic trick with numbers!

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Calculating Factorial Using Previous Value

Wondering how to calculate factorial more efficiently? The factorial discussed adiscussed above shows something interesting: a number's factorial is found by multiplying it with the factorial of the number before it.

In other words, n! = n x (n-1)!

For example, 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 8 x 7!

Again, it also helps to calculate the factorial of a number if the factorial of its successor is known.

The formula will be: n! = (n+1)!/(n+1)

For example, 4! = 5!/5 = (5 x 4 x 3 x 2 x 1)/5 = 4 x 3 x 2 x 1 = 24

How to Calculate Factorial With Your Fingers

If you wish to know how to do a factorial without weighing it down on a calculator, use your fingers. Example: for 4!, just put four fingers up and count backward. 4 × 3 = 12, 12 × 2 = 24, and finally, 24 times by 1 gives you 24. Quite interesting, isn't it? Even bigger numbers can be solved this way step by step. So next time someone asks you what a factorial number is, show them your finger trick.

Factorial of One

The factorial of 1 is expressed as 1! =1 as per the factorial formula. It can also be explained differently. In permutations, the number of ways of arranging ‘n’ different things can be obtained by the value of n! Since 1 can be arranged in only one way, it explains the value of factorial one as 1.

Factorial of Zero

The factorial of zero can be calculated by the formula: n! = (n+1)!/(n+1)

We can write, 0! = 1!/1 = 1/1  = 1

So, the factorial of zero is 1, which is the same as the factorial of 1.

Factorial of Negative Numbers

The factorial of negative numbers is undefined . This can be explained in the following way.

We know that n! = (n+1)!/(n+1)

If we want to find the value of (1)! The calculation will be:

(-1)! = 0!/0 = 1/0, but anything divided by zero is undefined.

So, the factorial of a negative number can’t be defined.

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Factorials in Statistics

There are many factorial properties, factorial in statistics is one of them. Factorials are used in probability while calculating the possible ways of arranging a certain number of objects.

In permutations , the different ways several things can be arranged in a definite order can be calculated as: n P r = n! / (n - r)!

Where n is the total number of objects and r is the number of objects chosen at a time.

For example, let’s take this situation: In how many ways the 6 persons a, b, c, d, e, and f can achieve the position of 1 st , 2 nd , and 3 rd ?

The formula for the solution to this problem is:

6! / (6-3)!

= 6! / 3!

= (6 x 5 x 4 x 3 x 2 x 1)/ (3 x 2 x 1) = 6 x 5 x 4 = 120

In combinations, we find the grouping of some things without any definite order. The formula is: n C r = n! / [ (n - r)! r!]

Where n is the total number of objects and r is the number of objects chosen at a time.

For example, let’s take this problem: In how many ways can 3 prizes be distributed to 8 people?

The calculation for the solution to this problem is:

= (8!) / [ 3! (8 - 3)!] = 8! / (3! 5!)

= (8 x 7 x 6 × 5!) / [(3 × 2 × 1) 5!]

= 8 x 7 x 6/ 3 x 2 x 1

=8 x 7

= 56 ways

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Factorial Values F or Numbers 1 to 10

Students often require the values of the factorials from 1 to 10 to make quick calculations. A list of factorial values from 1 to 10 is given below as a quick reference.

Factorials Solved Examples

1. Find the value of the expression 7! - 5!

Solution:

7! - 5! = (7 x 6 x 5!) -5!

7! - 5! = 5!(7 x 6 - 1)

7! - 5! = 5! (42 -1)

7! - 5! = 5! x 41

7! - 5! = 5 x 4 x 3 x 2 x 1 x 41

7! - 5! = 4920

2. Find the value of 16! / (13! × 5!)

Solution:

16! / (13! × 5!) = (16 × 15 × 14 × 13!) / (13! × 5!)

16! / (13! × 5!)  = (16 × 15 × 14) / 5!

16! / (13! × 5!)  = (16 × 15 × 14) / (5 x 4 × 3 × 2 × 1)

16! / (13! × 5!)  = (16 × 15 × 14) / (15 × 8)

16! / (13! × 5!)  = (16 x 14) / 8

16! / (13! × 5!) =  2 x 14

16! / (13! × 5!) = 28

3. In how many ways can 3 digits be arranged to form different 3-digit numbers in which no digit is repeated?

Solution:

Number of digits = 3

6 We have to arrange these digits to form a 3-digit number.

The number of ways to arrange these digits to form a 3-digit number is 3!

3! = 3 × 2 × 1 = 6

Thus, there are 6 ways 3 digits can be arranged to form different 3-digit numbers.

The factorial is an important concept in mathematics that helps solve problems related to permutations and combinations, binomials, and algebra. It provides effective ways to analyze and organize data. Understanding factorials aids in quick mathematical computations and improves problem-solving skills.