Factorial

 

 

For any positive integer n,

               n! = n (n - 1)(n - 2)(n - 3) ... 3.2.1

 

   n! is read as n factorial

 

Useful rule of factorials:

                     n! = n (n - 1)!

 

     Why is this so?

                  From definition,

                          n! = n (n - 1)(n - 2)(n - 3) ... 3.2.1

                n (n - 1)! = n [(n - 1)(n - 2)(n - 3) ... 3.2.1]

     From here, both are obviously the same.

 

     Similarly, (n + 1)! = (n + 1) n!

 

Examples:     3! = 3.2.1 = 6

                      4! = 4.3.2.1 = 24

                      5! = 5.4.3.2.1 = 120

 

 

Express the following using factorials only:

 

          ( multiply by )

 

3.5.7.9.11.13.15.17.19

 

    Add in the even numbers:

              

 

Note that in the second line, the numerator is now 19! and the denominator has common factor 2. Also note that 2! = 2.