Permutations
The number of ways you can change the order of a set of things is called
the number of PERMUTATIONS of that set of things. For example, how
many different ways can you arrange the letters in the
word
WHO
Answer: WHO WOH HWO HOW OHW OWH = 6 ways
1 2 3 4 5 6
Each different letter arrangement is called a permutation of the
word "WHO".
How about the word "STOP"? Well, here they are:
STOP STPO SOTP SOTP SPTO SPOT < starts with "S"
TSOP TSPO TOSP TOPS TPSO TPOS < starts with "T"
OSTP OSPT OTSP OTPS OPST OPTS < starts with "O"
PSTO PSOT PTSO PTOS POST POTS < starts with "P"
There are 24 ways to order the letters in "STOP".
Is there a general rule here? Fortunately, yes. Here's the rule for
"STOP":
 There are 4 ways to pick the first letter.
 After you pick the first letter there are 3 ways to pick the second
letter.
 After you pick the first 2 letters, there are 2 ways to pick the third
letter.
 After picking the first 3 letters, there is only 1 letter left to
pick.
So the number of ways to order the letters in "STOP" is 4 x 3 x 2 x 1 = 24
ways!
FACTORIALS
When you multiply a whole number by all the whole numbers below it, that
is called the FACTORIAL of that number.
Factorials are used to compute permutations. In the previous example, we
saw that 4 letters could be permuted 4 x 3 x 2 x 1 ways , or 24 ways.
Six things can be permuted 6 x 5 x 4 x 3 x 2 x 1 ways or 720 ways. We
write it in shorthand with an exclamation point, like this: 6! You can
see from the table below that factorials get very large in a hurry:
1! = = 1
2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880
The number of permutations of n
things is n! 
There is one more thing to know about factorials. When you divide one
factorial by another, then a lot of the numbers on the low end cancel out.
For instance:
12! 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
 =  = 12 x 11
10! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
REPEATED LETTERS
There is a special case when the things you are ordering are not all
different. This is the case when you are permuting the letters in the word
"MOON". Here are the permutations:
MOON MONO MNOO
OMON OMNO OOMN
OONM ONMO ONOM
NOOM NOMO NMOO
Why are there only 12, instead of 4 x 3 x 2 x 1 = 24 ways? Because you
can't tell the "O"'s apart, so their order doesn't matter.
Suppose one of the "O"s was capital and the other one was small. Then you
would have "MOoN" and "MoON". But our "O"s are alike and we can't tell the
difference so we have "MOON" and "MOON", which are the same. We count them
once. Since there are 2 "O"s, there are 2! = 2 x 1 = 2 ways of ordering
them. So the number of permutations of letters in the word MOON is:
4! 4 x 3 x 2 x 1
 =  = 4 x 3 = 12 ways
2! 2 x 1
For the word "SASS", there are 4 letters and 3 "S"s so there are:
4! = 4 x 3 x 2 x 1
  = 4 ways of permuting the letters
3! 3 x 2 x 1 ( = SASS SSSA ASSS SSAS)
The number of permutations of n things
where r of them are the same is :
