Factorial: Determines the number of different ORDERS in which one can arrange or place set of items

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

A factorial indicates the multiplication of consecutive natural numbers.




Example 1:
The senior choir has rehearsed five songs for an upcoming assembly.
In how many different orders can the choir perform the songs?

Solution:

There are five ways to choose the first song, four ways to choose the second, three ways to choose the third, two ways to choose the fourth, and only one way to choose the final song. Using the fundamental counting principle, the total number of different ways is

5x4x3x2x1 = 5!
=120
The choir can sing the five songs in 120 different orders.

Example 2: Evaluate



=84

Example 3:


Example 4:




A Permutation of 'n' distinct items is an arrangement of all the items in a definate order.

A permutation of 'n' distinct items taken 'r' at a time is an arrangement of 'r' of the 'n' items in a definate order.


A rearrangement of the elements of a set.

Example 1: The word bookkeeper is unusual in that it has three consecutive double letters. How many permutations are there of the letters in bookkepper?

Solution:
If each letter were different, there would be 10! permutations, but there are two os, two ks, and there es. You must divide by 2! twice to allow for the duplication of the os and ks and then divide by 3! to allow for the three es:




=151 200

There are 151 200 permutations of the letters in bookkeeper.


The following are all the permutations of DOLE:

DOLE,DOEL,DLOE,DLEO,DEOL,DELO
ODLE,ODEL,OLDE,OLED,OEDL,OELD
LODE,LOED,LDOE,LDEO,LEOD,LEDO
EOLD,EODL,ELOD,ELDO,EDOL,EDLO

There are 24 permutations of the letters in DOLE. This number matches what you would calculate using 4P4 = 4!