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12 1 The Fundamental Counting Principal & Permutations Revised
 

12 1 The Fundamental Counting Principal & Permutations Revised

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    12 1 The Fundamental Counting Principal & Permutations Revised 12 1 The Fundamental Counting Principal & Permutations Revised Presentation Transcript

    • 12.1 The Fundamental Counting Principal & Permutations
    • The Fundamental Counting Principal
      • If you have 2 events: 1 event can occur m ways and another event can occur n ways, then the number of ways that both can occur is m*n
      • Event 1 = 4 types of meats
      • Event 2 = 3 types of bread
      • How many diff types of sandwiches can you make?
      • 4*3 = 12
    • 3 or more events:
      • 3 events can occur m, n, & p ways, then the number of ways all three can occur is m*n*p
      • 4 meats
      • 3 cheeses
      • 3 breads
      • How many different sandwiches can you make?
      • 4*3*3 = 36 sandwiches
      • At a restaurant at Cedar Point, you have the choice of 8 different entrees, 2 different salads, 12 different drinks, & 6 different deserts.
      • How many different dinners (one choice of each) can you choose?
      • 8*2*12*6=
      • 1152 different dinners
    • Fund. Counting Principal with repetition
      • Ohio Licenses plates have 3 #’s followed by 3 letters.
      • 1. How many different licenses plates are possible if digits and letters can be repeated?
      • There are 10 choices for digits and 26 choices for letters.
      • 10*10*10*26*26*26=
      • 17,576,000 different plates
    • How many plates are possible if digits and numbers cannot be repeated?
      • There are still 10 choices for the 1 st digit but only 9 choices for the 2 nd , and 8 for the 3 rd .
      • For the letters, there are 26 for the first, but only 25 for the 2 nd and 24 for the 3 rd .
      • 10*9*8*26*25*24=
      • 11,232,000 plates
    • Phone numbers
      • How many different 7 digit phone numbers are possible if the 1 st digit cannot be a 0 or 1?
      • 8*10*10*10*10*10*10=
      • 8,000,000 different numbers
    • Testing
      • A multiple choice test has 10 questions with 4 answers each. How many ways can you complete the test?
      • 4*4*4*4*4*4*4*4*4*4 = 4 10 =
      • 1,048,576
    • Using Permutations
      • An ordering of n objects is a permutation of the objects .
    • There are 6 permutations of the letters A, B, &C
      • ABC
      • ACB
      • BAC
      • BCA
      • CAB
      • CBA
      You can use the Fund. Counting Principal to determine the number of permutations of n objects. Like this ABC. There are 3 choices for 1 st # 2 choices for 2 nd # 1 choice for 3 rd . 3*2*1 = 6 ways to arrange the letters
    • In general, the # of permutations of n objects is:
      • n! = n*(n-1)*(n-2)* …
    • 12 skiers…
      • How many different ways can 12 skiers in the Olympic finals finish the competition? (if there are no ties)
      • 12! = 12*11*10*9*8*7*6*5*4*3*2*1 =
      • 479,001,600 different ways
    • Factorial with a calculator:
      • Hit math then over, over, over.
      • Option 4
    • Back to the finals in the Olympic skiing competition.
      • How many different ways can 3 of the skiers finish 1 st , 2 nd , & 3 rd (gold, silver, bronze)
      • Any of the 12 skiers can finish 1 st , the any of the remaining 11 can finish 2 nd , and any of the remaining 10 can finish 3 rd .
      • So the number of ways the skiers can win the medals is
      • 12*11*10 = 1320
    • Permutation of n objects taken r at a time
      • n P r =
    • Back to the last problem with the skiers
      • It can be set up as the number of permutations of 12 objects taken 3 at a time.
      • 12 P 3 = 12! = 12! = (12-3)! 9!
      • 12*11*10*9*8*7*6*5*4*3*2*1 = 9*8*7*6*5*4*3*2*1
      • 12*11*10 = 1320
    • 10 colleges, you want to visit all or some.
      • How many ways can you visit
      • 6 of them:
      • Permutation of 10 objects taken 6 at a time:
      • 10 P 6 = 10!/(10-6)! = 10!/4! =
      • 3,628,800/24 = 151,200
    • How many ways can you visit all 10 of them:
      • 10 P 10 =
      • 10!/(10-10)! =
      • 10!/0!=
      • 10! = ( 0! By definition = 1)
      • 3,628,800