Combinations

And Problems with Combinations

Discussion

• Combinations, like permutations are related the arranging objects from a set of objects.
• Unlike permutations, we don't care about the order.
• A combination is a distinct set group of objects without regard to their assignment.
• Example 1

  A club has 10 members, how many ways can they select a 
  president, vice president and treasurer.
  
  This is a permutation, since it matters who is president.
  
     10!/7!
  
  10 * 9 * 8 =  720
  
  How many ways can they select a council of three people?
  
  This is a combination, since the first councilor is no different 
  from the second and the third.
      

• To compute combinations, you use the following formula

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

• So in the above example

      There are 10 people, and three seats
                10!
      10C3 = ----------- 
             (10-3)!3!
  
  	= 10*9*8 / 3*2 = 10*3*4 = 120
  
      

• Example 2, #22, Page 728

  Since we don't care about the order of the toppings this is a combination.
  
  20C3 = 20*19*18/3*2 = 1710
      

• Example 3, #28

  Once again, we don't care about the order of the plants.
  
  24C20 = 24!/(4!*20!) 
                              = 24*23*22*21/4*3*2
                              = 23*22*21
                              = 10626 
      

Problems with Combinations.

• Remember the basics of probability
• P(E) = n(event happening)/ n (events)
• Example 4, #10, page 734

  This is a combination problem, we don't care about the order, just the color.
  
         3C2
  P(E) = ___     
         6C2
  
       = 1/5
      

• Example 5, #12,

         4C2     3
  P(E) = ___  = __
         8C2    14
      

• Example 6, #18

  
         26C5    289
  P(E) = ___ = ____ 
         52C5   9996
      

Homework

Please do 9-19 odds page 734