# Factorial of any number

Introduction to factorial of any number :

In general, factorial is always defined for a positive real number. Let the positive real number be n. Then, the symbolic representation of factorial of any real number is given by " n ! ." The factorial of any number "n" can be defined as the product of all positive numbers less than or equal to n.

For example : 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1Note: 0! is an exception and it is equal to 1.The main utilisation of factorial is in the field of algebra and mathematical analysis. Taking the particular case of algebra, the simplest example is arrangement of books on a shelf. It means, there are n! ways of arrangement of n distinct books on the shelf (that is, permutations of the set of books). It was Christian Kramp, who introduced the notation n! in the year 1808.

The most general definition of factorial is :n! = { 1, when n = 0{ (n -1)! × n , when n > 0Points to Remember in Factorial Equation:The equation involving factorial numbers is known as factorial equation.example : (n - 2)! = (n - 2) × (n - 3)Some important points to remember:(1)0! = 1.(2) When n is negative or a fraction, n! is not defined.

Solved Problems on Factorial Equation:(1) 1/ (8!) + 1/ (9!) = n/(10!)

Solution: 1/(8!) + 1/ (9 × 8!)

= n/ (10 × 9 × 8!)Or,1 + 1/9

= n/(10 × 9)Or, 1 + 1/ 9

= n/ 90Or, 10/9

= n/90Or,

n = (10 × 90) / 9Or,

n =10 × 10 = 100

(2) Find the value of n if n! / (n -2)! = 12 ?

Solution: (n) × (n -1) × (n – 2)!/(n – 2)!

= 12Or, (n) × (n -1)

= 12Or, n2 - n -12

= 0

Or, n2 -4n + 3n - 12 = 0

Or, n ( n – 4) + 3 (n – 4) = 0

Or, (n – 4) (n + 3) = 0Or, n = 4 ( since, n = (- 3) is not valid as “n” cannot be negative)