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":
  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?



WORD
LENGTH
EXAMPLEPERMUTATIONSRESULT
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! Here it 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: