Factorial

Factorial

Factorial of a number

Definition: 

   The product of all consecutive positive integers from 1 to n, is called Factorial of n.

The factorial of a non negative integer is denoted by  n!.

  i.e., n!= 1.2.3...(n-2)(n-1)×n.

              = n(n-1)...3×2×1. , for all integers n greater than or equal to 1.

Factorial of 0

The factorial of zero is defined to be 1.

 0!=1.

Significance of n! 

We know that, the no. of permutations in choosing r elements from n elements is= n(n-1)(n-2)...(n-r+1).

Now, choosing  r=n, i.e., no. of permutations in taking n elements out of n elements is= n(n-1)(n-2)...3×2×1 = n!.

Significance of 0!

Simply, we can say that, the no. of ways to choose  0 element from the empty set is: 

0!/(0!×0!)=1.

 More generally, we can say that, the no. of ways to choose all the n elements among the set of n elements is:

   n!/(n!×0!)= 1.

Important property of factorial of n is: n!= n×(n-1)!.

 Note:

 ( a±b)! ≠ (a! ± b!).

  (a×b)! ≠ (a! × b!).

Some factorials:

   0! = 1.

    1!=1.

    2!=2.

    3!=6.

    4!=24.

    5!=120.

    6!=720.

    7!=50400.

     8!=40320.

     9!=362880.

     10!=3628800.

     11!=39916800.

      12!=479001600.

 Factorial of a non integer:

The factorial of a non integer can be defined using the gamma function such that, n!=π(n+1).

  Here one question may arise that, why the factorial of a negative integer does not exist?

   The simple answer is that, the factorial of a negative integer is not defind.

How does our computers calculate the value of factorial of a number?

     Mathematically  calculating  the factorial  of a number is easy. We  just multiply a bunch of numbers together. However, simple as it may seem, most computers don’t find the answer by just multiplying.  There are certain limitations.
     Factorials are always integer numbers because it is  the result of multiplying integers together.
     Modern computers covert our everyday numbers into binary, before they do any calculation. Older computers are limited to working with 32 binary digits, or bits. which translate to a maximum of 2,147,483,647 in decimal. For the newer 64-bit computers, it can store an integer in    decimal value upto 2⁶³ -1.
Which is a big number. But if you look at the table, you will see that a 32-bit computer can only calculate up to 12! and   a 64-bit computer gives accurate value upto 20!. Beyond these boundaries, most common computers  provides an approximate answer.

     

  Alternating factorial:
 The alternating factorial of a positive integer "n" is the absolute value of the  alternating sum of the factorials of the positive integers 1,2,3,...,n.
  Mathematically, we denote alternating factorial of 'n' as , af(n) and define as,
  af(n)= n! - af(n-1), [using recurrence relation]. Here, af(1)=1.
 As for example, af(4)= 4! -3! +2! - 1! = 19.

Exponential factorial:

The exponential factorial of a positive integer n is defined as the raising powers of the integers n-1, n-2, n-3,... exponentially (i.e., ((nⁿ⁻¹)ⁿ⁻²)ⁿ⁻³...  ).

 Using recurrence relation exponential factorial is defined as follows:

 aₙ = nᵃₙ_₁, where aₒ=1.

 As for example, 9 is an exponential factorial(as, 9=(3²)¹ ). 

    

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