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THE FUNDAMENTAL COUNTING PRINCIPLE & PERMUTATIONS Computer Science, Statistics and Probability all involve counting techniques which are a branch of mathematics called combinatorics (ways to combine things). We'll be introducing this topic in this section. For dinner you have the following choices: ENTREES soup MAINS salad chicken prawns hamburger DESSERTS How many different combinations of meals could you make? icecream We'll build a tree diagram to show all of the choices. Notice the number of choices at each branch 2 choices 3 choices 2 choices We ended up with 12 possibilities soup, chicken, ice cream soup, chicken, 2 3 2 = 12 prawns soup, prawns, ice cream soup, prawns, soup, hamburger, ice cream soup, hamburger, salad, chicken, ice cream salad, chicken, prawns salad, prawns, ice cream salad, prawns, Now to get all possible choices we follow each path. salad, hamburger, ice cream salad, hamburger, THE FUNDAMENTAL COUNTING PRINCIPLE & PERMUTATIONS ESSENTIAL QUESTION How is the counting principle applied to determine outcomes? Multiplication Principle of Counting If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the task of making these selections can be done in different ways. pqr If we have 6 different shirts, 4 different pants, 5 different pairs of socks and 3 different pairs of shoes, how many different outfits could we wear? 6 4 5 3 = 360 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 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 FUNDAMENTAL 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 HOW MANY PLATES ARE POSSIBLE IF DIGITS AND NUMBERS CANNOT BE REPEATED? There are still 10 choices for the 1st digit but only 9 choices for the 2nd, and 8 for the 3rd. For the letters, there are 26 for the first, but only 25 for the 2nd and 24 for the 3rd. 10*9*8*26*25*24= 11,232,000 plates PHONE NUMBERS How many different 7 digit phone numbers are possible if the 1st 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 = 410 = 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 Fundamental Counting Principle to determine the number of permutations of n objects. Like this ABC. There are 3 choices for 1st # 2 choices for 2nd # 1 choice for 3rd. 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