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Factorials

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Factorial

In mathematics, factorials are discovered to find out the factorial of a number. Factorial conception is used in many mathematical impressions, such as geometry, number theory, permutations, probability, and combinations etc. The factorial function was discovered by Daniel Bernoulli. Factorials are multiplication of a number less than equal to ‘n’ till 1. For example, factorial of 6 is, 6 × 5 ×  4 × 3 × 2 × 1 that is 720. 

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What is Factorial

Factorial is a very significant function. In this article, we will discuss what is the meaning of factorials? What is its notation and formula? In this, we will study factorials of negative, decimal factorials, half factorial and double factorial. The main purpose of this article will be how to calculate a factorial with examples? Also we will explain the properties and application of factorial.

Factorials Notation and Formula

In factorials, factorial is represented by the exclamation mark “!” If we have “n!” we read these symbols as ‘n factorial’ where n is a number.  

Formula for finding the factorial of the number

    \[n\]

is.

    \[n! = \dfrac{{(n + 1)!}}{{(n + 1)}}\]

 

If we have 6!

So what is the actual meaning of this? It is look like complicated, for simplify this 

    \[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\]

 

That means 

    \[6! = 720\]

 

 Specific Factorials

So let us first understand how the factorial work we know that,

    \[\begin{array}{l}1! = 1\\2! = 2 \times 1!\\3! = 3 \times 2!\\4! = 4 \times 3!\end{array}\]

 

Let us simply see how actually you can make this happen,

 Let see

For example, how did we find 3! 

3! Is found by dividing 4! By 4

    \[3! = \dfrac{{4!}}{4} = \dfrac{{4 \times 3 \times 2 \times 1}}{4} = 6\]

 

    \[\begin{array}{l}3! = 3 \times 2!\\4! = 4 \times 3!\end{array}\]

 

Similarly, 

How did we find 2!

2! Is found by dividing 3! By 3

    \[\begin{array}{l}2! = \dfrac{{3!}}{3} = \dfrac{{3 \times 2 \times 1}}{3} = 2\\3! = 3 \times 2!\\4! = 4 \times 3!\end{array}\]

 

If we want to find out the value of 5!

5! Is found by dividing 6! By 6 

    \[5! = \dfrac{{6!}}{6} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{6} = 120\]

 

Similarly

Find the value of 10! Dividing 11! By 11

10! = 11×10×9×8×7×6×5×4×3×2×111 = 3628800

Now 0! Will be 

To find out the value of 0! 

First we know that 1! = 1

If we notice for 4! =5!5, for 3! =4!4, for 2! =3!3, 1! =2!2, 

The same method goes for 0! So it will be 1!1 = 1

Factorials of negative Number

Now we discuss the factorials of negative numbers. Suppose we want to find out the negative number, say (-1).

We know the formula to find the value of 

    \[n! = \dfrac{{(n + 1)1}}{{(n + 1)}}......A\]

 

Put the value of n in this formula.

    \[ - 1 = \dfrac{{( - 1 + 1)!}}{{( - 1 + 1)}} = \dfrac{{0!}}{0} = \dfrac{{anything}}{0} = notdefine\]

 

Similarly

For n = – 2

Put the value in equation A.  

    \[ - 2 = \dfrac{{( - 2 + 1)!}}{{( - 2 + 1)}} = \dfrac{{ - 1!}}{{ - 1}} = notdefine\]

 

Therefore, the factorial of negative number is not define

Decimal Factorials

We have factorials for decimal numbers such as 0.5 or – 3.217.

For this factorial we need to use the gamma function.

Half Factorial

These are the some half factorial function 

    \[\begin{array}{l}\left( { - \dfrac{1}{2}} \right)! = \sqrt \pi \\\left( {\dfrac{1}{2}} \right)! = \left( {\dfrac{1}{2}} \right)\sqrt \pi \\\left( {\dfrac{3}{2}} \right)! = \left( {\dfrac{3}{2}} \right)\sqrt \pi \\\left( {\dfrac{5}{2}} \right)! = \left( {\dfrac{{15}}{8}} \right)\sqrt \pi \end{array}\]

 

Double Factorial!!

Double factorial is represented by double exclamation mark “!!” for finding the value of double factorial? Double factorial is like simple factorial. But you skip every alternate value.

  • 5!! = 5 3 1 = 15
  • 8!! = 8 6 4 × 2 = 384 
  • 9!! = 9 7 5 3 1 = 954

How to calculate a factorial

If we have n where n is any number then to calculate its factorials we will write the number then multiply it by its one less number, then again multiply it by its one less number. We will do this until we get 1 number. 1 is the last number of products of factorial.

    \[n! = n \times (n - 1) \times (n - 2) \times (n - 3)....3 \times 2 \times 1\]

 

For example 1:

What is the 8!

8! Is 8 × (8 – 1) × (8 – 2) × (8 – 3) × (8 – 4) × (8 – 5) × (8 – 6) × (8 – 7)

8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 

So 8! Is 40,320

Example 2:

Evaluate the value of 9!

9! Is 9 ( 9 – 1) × ( 9 -2 ) ( 9 – 3 ) ( 9 – 4 )  × ( 9 – 5 ) ( 9 – 6 )  × ( 9 – 7) ( 9 – 8)

9 × 8 × 7 6 × 5 4 3 2 1

So 9! Is 362880

Application of Factorial

The factorial function will be used to find out how many variant ways to choose an item from a group.

 For example:

If we have 3 books, we can order them in 6 different ways, but if we have 4 books, there are 24 ways you can order them.

3! = 3 × 2 × 1

6

4! = 4 × 3 × 2 × 1

24

Factorial is helpful in advanced mathematics, for series and sequences. 

Conclusion

Meaning of factorial is the product of any number. In this chapter we learned that factorial is multiplication of numbers and we learned that factorial is represented by “!” In this we studied the value of 0! Is 1.In this we studied the  factorial of negative. The number is not defined. There are many different things, factorials help us to choose the things.   

Solved Examples

  1. Find the value of 5!

5! = 5 × 4 × 3 × 2 × 1

120

  1. Evaluate

        \[\dfrac{{n!}}{{(n + r)!}}\]

    when n = 4, r = 2

Put the value of n and r in equation 

4!(4+2) = 4 !6 × 5 × 4 ! = 130

  1. Evaluate 4! – 3!

4! = 4 × 3 × 2 × 1

24

3! = 3 × 2 × 1

6 

24 – 6

18

  1. Find 10! + 7!

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

3628800

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

5040

3628800 + 5040 = 3633840

  1. Find the value of  7!6! 

7!6! = 7 ×6!6! = 7

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FAQs

1. What is the value of 0!? Prove.

Ans: We know the factorial formula n! =  n+1!( n+1) 

Put the value in formula 0! = 0+1!( 0+1) = 1

Hence prove the value of 0! Is 1.

2. Simplify the equation (5! + 2!).  

Ans: 5! = 5 × 4 × 3 × 2 × 1

120

2! = 2 × 1

2 = 120 + 2 = 122

3. Solve the equation if n!n+1!  if n = 4.

Ans: n!(n+1)!  Put the value of n in this equation 

4!(4+1)! = 4!5! = 4! x 5 × 4! = 15

4. Can you prove that 8! = 3!  + 5!?

Ans: 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

40320

3! = 3 × 2 × 1

6

5! = 5 × 4 × 3 × 2 × 1

120

40320 126

5. What is the value of 11!?

Ans: 11 = 11 x 10 x 9 x 8× 7 × 6 x 5 × 3 × 2 x 1

10395

References

https://www.purplemath.com/modules/factorial.htm

https://www.freecodecamp.org/news/what-is-a-factorial/

https://mathworld.wolfram.com/Factorial.html

https://byjus.com/maths/factorial/

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