The Fundamental Counting Principle & Permutations

Transcription

1 The Fundamental Counting Principle & Permutations POD: You have 7 boxes and 10 balls. You put the balls into the boxes. How many boxes have more than one ball?

2 Why do you use a fundamental counting principal? What operation do you use for fundamental counting principals? What is a permutation? What is the formula for n P r? What is the formula for permutations with repetition?

3 The Fundamental Counting Principle 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 different types of sandwiches can you make? 4*3 = 12

4 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

5 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

6 Fund. Counting Principle 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

7 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

8 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

9 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

10 Using Permutations An ordering of n objects is a permutation of the objects.

11 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

12 In general, the # of permutations of n objects is: n! = n*(n-1)*(n-2)*

13 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

14 Factorial with a calculator: Hit math then over, over, over. Option 4

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16 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

17 Introduction to Permutations y/v/permutations

18 Permutation of n objects taken r at a time n P r =

19 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 = 12*11*10 = *8*7*6*5*4*3*2*1

20 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

21 How many ways can you visit all 10 of them: 10 P 10 = ( 0! By definition = 1) 10!/(10-10)! = 10!/0!= 10! = 3,628,800

22 So far in our problems, we have used distinct objects. If some of the objects are repeated, then some of the permutations are not distinguishable. There are 6 ways to order the letters M,O,M MOM, OMM, MMO MOM, OMM, MMO Only 3 are distinguishable. 3!/2! = 6/2 = 3

23 Permutations with Repetition The number of DISTINGUISHABLE permutations of n objects where one object is repeated q 1 times, another is repeated q 2 times, and so on :

24 Find the number of distinguishable permutations of the letters: OHIO : 4 letters with 0 repeated 2 times 4! 2! = 24 2 = 12 MISSISSIPPI : 11 letters with I repeated 4 times, S repeated 4 times, P repeated 2 times 11! = 4!*4!*2! 39,916,800 = 24*24*2 34,650

25 Find the number of distinguishable permutations of the letters: SUMMER : 360 WATERFALL : 90,720

26 A dog has 8 puppies, 3 male and 5 female. How many birth orders are possible 8!/(3!*5!) = 56

27 Why do you use a fundamental counting principal? To count the number of possibilities of the given conditions. What operation do you use for fundamental counting principals? Multiplication What is a permutation? An ordering of objects. What is the formula for n P r? What is the formula for permutations with repetition?

28 Assignment worksheet