The Fundamental Counting Principle

• The Fundamental Counting Principle states: If you can choose one item from a group of M items and a second item from a group of N items, then there are a total of MxN two item combinations.
• Or The number of ways in which a series of successive things can occur is found by multiplying the number of ways in which each thing can occur
• Examples:

  2: page 612
      9 x 3 = 27
  4:
      5 x 6 = 30
  6:
      26 letters, 9 digits
      26 x 9 =  234
  8:
      3 sizes, 4 crusts, 6 toppings
      3 x 4 x 6 = 72
  12:
      3 x 2 x 4 = 24
  
  14:
      9 colors x 2 air x 2 sunroof x 2 transmission x 2 brakes = 144
  
  20:
      2 x 26 x 26 x 26 = 35,152
      

Permutations

• We can apply the fundamental counting principle to multiple selections from the same group.
• In this case, the order in which we place the order matters.
• If we have three groups, to perform, how many ways can we arrange the schedule?

  We can choose 1 of 3 for the first slot (3)
  We can choose 1 of 2 for the second slot (2)
  We can choose 1 of 1 for the third slot (1)
     3 x 2 x 1 = 6
      

• Example: Question 4 page 619

  8 x 7 x 6 x 5 x 4 x 3 x 2 = 40,320
      

• Example

  10:
     1 choice for the first slot
     3 choices for the second slot
     2 choices for the third slot
     1 choice for the fourth slot
     1 choice for the fifth slot
  
     1 x 3 x 2 x 1 x 1 = 6
      

• These types of problems always involve n x (n-1) x (n-2) x ... x 3 x 2 x 1
• We have a mathematical short hand for this.
  • It is called n factorial
  • written n!
  • 0! = 1
  • 1! = 1
  • 2! = 2 x 1 = 2
  • 3! = 3 x 2 x 1 = 6
  • Notice, for n > 1 , n!= n x (n-1)!
• _nP_r = n!/(n-r)!
• Your calculator should have a n! and a _nP_r button learn how to use these.
• Example:

  5P3 = 5!/3! = 5x4x3!/3! = 20
      

• Any questions 13 through 40?
• Example:

  42:
      10P3 = 10!/3! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3!/3!
                                 = 604,800
      

• If some items are identical, where order is not noticeable.
• From a set of n items, where p items are identical, q are identical and r are identical then there are

                    n!
  	     ------------
               p! x q! x r!
      

• Example

  40
    SCIENCE  
    7 total letters
    E is repeated twice
    C is repeated twice
    S,I,N repeated once.
  
    7!/(2!x2!x1!x1!x1!) = 7!/4 = 1,260
      

Homework

• Page 612: 1-22 odd
• Page 619: 1- 52 odd