The Counting Principle and Permutations

• Factorial: a function of a non-negative integer
  • written n! (1!, 2!, ....)
    • This notation was first used in 1808
    • By a French Mathematician Kramp.
  • 0! = 1 by definition
  • 1! = 1 by definition
  • n! = n× (n-1)! by definition.
  • So 2! = 2× 1 = 2
  • 3! = 3×2! = 3*2 = 6
  • 4! = 4×3! = 24
  • 5! = 5 × 4! = 120
  • 6! = 6×5! = 720
  • Factorial gets big fast.
  • I have five dice, how many ways can I arrange them on the shelf?
  • I have 7 pictures, how many ways can I arrange them on the wall?
  • I plan to play 12 songs at a party, how many different ways can I do this?
  • I have 20 different elephant jokes, how many different ways are there write them down on a sheet of paper.
  • Notice a pattern?
• Sometimes we don't use all of the items.
  • What If I only have space to display two of my five dice? How many ways can I do that?
  • How about only three of my seven pictures?
  • How about only four elephant jokes?
• In all of these cases, we were selecting a subset (perhaps the entire universal set) of items.
• In all of the cases the order was important.
• Such an operation is called a permutation
• A permutation is any ordered arrangement of a given set of objects.
• We can calculate the number of permutations with the following formula

              n!
     nPr =  ------ 
            (n-r)!
     

• Does this make sense?
  • Consider three of the seven pictures.

                7!      7×6×5×4!
       7P3 =  ------  = --------
              (7-3)!          4!
    
           

• Examples:
  • An ATM Machine requires a 4 digit code, using the digits 0-9. How many different cods are there
    • If repetition is not allowed
    • If repetition is allowed.
  • How many different license plates are possible if the plate contains three letters followed by 4 digits.
  • How many passwords are possible if a password consists
    • Seven lower case letters
    • Seven lower or upper case letters
    • Seven lower or upper case letters or digits.
    • Seven lower or upper case letters, digits or special symbols (15)
  • A club consists of ten members, how many different ways can they select a president, vice president treasurer and secretary
  • A DJ has 9 songs to play , Five are slow and 4 are fast. Each song is is to be performed once. How many ways can he play the songs if
    • The songs are plied in any order
    • The first and last song must be slow songs.
    • The first two are fast songs.
    • The songs alternate fast slow fast slow ...
  • A history test has 10 events to be matched to 10 dates. Each date can be matched to only one event. How many ways can this test be answered?
• If there are repeated elements, we divide by the number of repeats.
  • The number of distinct permutations of n objects where n₁ of the objects are identical, n₂ of the objects are identical, ... n_r of the objects are identical can be found using the formula

            n!
     ----------------
     n1!×n2!× ... ×nr!
           

  • How many different permutations are there of the letters in bob?

            n = 3, n1 = 2 = 3!/2! = 3
    
    	bbo bob obb
           

  • How many different permutations are there of the letters in mississippi?

              m = 1 i = 4 s = 4 p = 2
    
         11!
    -----------
    4!×4!×2!×1!