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Factorial Notation


In this section we introduce a very useful notation, called the factorial notation. This notation is very convenient for representing the product of first n natural numbers.
1, 2, 3, ...., n.

Definition
If n is a natural number, then n factorial, denoted by n! orLn is defined to be the product of 1 2 3 (n-1)n
That is n! = 1 2 3 (n-1) n

As a special case, we define zero factorial to be 0! = 1.

We now list values of n! for some values of n.0! = 1,
1! = 1,
2! = 1 2 = 2
3! = 1 2 3 = 6,
4! = 1 2 3 4 = 24
5! = 1 2 3 4 5 = 120,
6! = 1 2 3 4 5 6 = 720
We can also define n! recursively as follows:

Recursive Definition of Factorial
We define  
       0! = 1 and n! = n(n-1) ! for n 1
Thus 7! = 7 (6!) = 7 (720) = 5040,
and 8! = 8(7!) = 8 (5040) = 40320

Example
  1. Compute 4 ! + 3! Is 4! + 3! = 7!?
  2. Evaluate when (a) n = 5, r = 2, (b) n = 6, r = 3
Solution
  1. We have 4! + 3! = 24 + 6 =30.Also 7! = 7(6!) = 7 6 5! = 7 6 5 (4!) = 7 6 5(24) = 5040
    Therefore, 4! + 3! 7!
  2. (a) When n = 5, r = 2, then
    === = 20
    (b) When n = 6, r = 3, then
    === = 120

Properties of Factorials

  1. It is a natural number.
  2. If n 5, then n! ends in a zero.
  3. If n > 1 then n! is divisible by 2, 3, 4, ... , n.
  4. If n 1, then n! is divisible by r ! for 1 r n. In fact

    = n(n-1) (n-2) .... (r +1)
  5. Product of r ( 1) consecutive natural numbers can be written as quotient of two factorials.
Let the r consecutive natural numbers be m, m + 1, m + 2, ...... , m + r - 1.
Then m (m +1 ) (m + 2) .... (m + r -1 )  =
                        =.




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