Notes on Permutations

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 SOPT 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!

Do you see the pattern here? Here's the pattern:

WORD
LENGTH EXAMPLE PERMUTATIONS RESULT
1 A 1 = 1
2 AM 2x1 = 2
3 BOY 3x2x1 = 6
4 GIRL 4x3x2x1 = 24
5 TABLE 5x4x3x2x1 = 120
6 PIANOS 6x5x4x3x2x1 = 720
7 PICTURE 7x6x5x4x3x2x1 = 5040
8 PURCHASE 8x7x6x5x4x3x2x1 = 40320

Now, let's suppose you only want to choose a few
letters out of your word. For example, you only
want to choose 2 letters out of the word "TABLE".
Here are all the ways to pick them:

TA   TB   TL   TE   AT   AB   AL   AE   BT   BA
BL   BE   LT   LA   LB   LE   ET   EA   EB   EL

There are 20 pairs.
Is there a rule here too? Of course there is:
1. There are 5 ways to choose the first letter.
2. After you choose the first letter,
there are 4 ways to choose the second letter.
So, the number of 2 letter permutations
of the 5 letter word "TABLE" is 5 x 4 = 20

How about a general rule? Here it is:
If you have a word with "N" letters in it,
and you only want to pick a few letters from it, then:

Number of letters you want	Calculate:	Example: TABLE N = 5 letters
2	N x (N-1)	5 x 4 = 20
3	N x (N-1) x (N-2)	5 x 4 x 3 = 60
4	N x (N-1) x (N-2) x (N-3)	5 x 4 x 3 x 2 = 120

Using factorials, this is:
P = N! / (N - M)!
where M is the number of letters you are selecting.
For our example of 3 letters out of the word TABLE, this becomes:
P = 5! / (5-3)! = 120 / 2! = 120 / 2 = 60 ways.