Hands-on math!


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":
  1. There are 4 ways to pick the first letter.
  2. After you pick the first letter there are
    3 ways to pick the second letter.
  3. After you pick the first 2 letters, there are
    2 ways to pick the third letter.
  4. 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:
1A1 = 1
2AM 2x1 = 2
3BOY 3x2x1 = 6
4GIRL 4x3x2x1 = 24
5TABLE 5x4x3x2x1 = 120
6PIANOS 6x5x4x3x2x1 = 720
7PICTURE 7x6x5x4x3x2x1 = 5040
8PURCHASE 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:

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