Notes on Permutations

Permutations - 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. For the word STOP, we see 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! <=== "SIX FACTORIAL!

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 (the number of ways of reordering them) 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 = 132
        10!             10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

This happens when you are selecting only a subset of the permutations. For example, if you are selecting 2 letters out of a 4-letter word, then that is:

4!      4 X 3 X 2 X 1       
--- =  ------------------ =  4 X 3 = 12
2!              2 X 1

The general equation:
When you are selecting m out of n things, the number of permutations is n! / m! For example: How many 2-letter combinations can you make out of the word STOP?
Well, n = the number of letters in STOP = 4 and m =2,
so the number of permutations is 4! / 2! = 24 / 2 = 12 and here they are:

ST  SO  SP TS  TO  TP  OS  OT  OP  PS  PT  PO

Now, here is a problem for you:

6 kids go to an amusement park and they have a 'log flume ride' which has seats for 4 kids.
How many ways can you fill the seats in the log flume ride?

n = _____
m = _____

Number of ways = ____