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COUNTING PRINCIPLEPurpose: To reason quantitatively and abstractly with counting principle, permutation and combination.To look for and make use of structure.Outcome: Student will determine how to count strategically using counting principle, permutation and combination. They will analyze decisions and strategies using these and probability concepts. They will derive formulas for permutation and SS.Math.Content.HSS.CP.B.9(+) Use permutations and combinations to compute probabilities of compound events and solve problems. HYPERLINK "" CCSS.Math.Content.HSS.MD.B.7(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Door Lock Problem: Ms. Cao is shopping for a door lock. Which lock is the most secure? Why?Option 1:Requires a 5-digit code like 4-4-4-3-1. Each digit can be repeated.Option 2: Requires a 3-number code like 11-24-11. Each number can be repeated.Option 3:Requires a 3-letter code like A-B-C. Each letter can be repeated.Option 4: The code can be 3-6 digit long.Each digit can be repeated. 1.You are at your school cafeteria that allows you to choose a lunch meal from a set menu. You have two choices for the main course (a hamburger or pizza), two choices of a drink (orange juice, apple juice) and three choices of dessert (cookies, ice-cream, jello). How many different meal combos (one main course, one drink, and one dessert) can you select? Explain.2.An ice-cream shop offers 2 types of cones, 31 flavors of ice cream, and 12 toppings. How many different ice-cream cones can a customer order if each order only has one cone, one flavor and one topping?3.How many different ways can a license plate be formed a)if 7 letters are used and no letter can be repeated?b)if 4 letters followed by 3 digits are used, no letter or digit can be repeated?c)if 7 letters or digits are used, each letter or digit can be repeated, and the first character must be a number?4.A multiple-choice exam consists of 15 questions, each of which has 4 possible answers. If a student guesses on all the questions, how many different sets of answers are possible?Counting PrincipleIf event M can occur in m ways and event N can occur in n ways, then the event M followed by event N can occur in __________ ways.5. A restaurant offers 5 choices of appetizer, 10 choices of main meal and 4 choices of dessert. A customer can choose to eat just one course, or two different courses, or all three courses. Assuming all choices are available, how many different possible meals does the restaurant offer? Write on a separate sheet and attach to this packet.PermutationsLinear Permutations with repetitions:How many different codes are possible for this “combination” lock?2.You are setting a password for your email account. Which of the following sets of criteria makes for a safer password?Criteria 1:The password must have six characters with only letters (A-Z) or digits (0-9).It is not case-sensitive.The first character must be a letter.The characters can be repeated. Criteria 2:The password must have four characters with only letters (A-Z) or digits (0-9).It is case-sensitive.One of the characters must be a number.The characters can be repeated. Linear Permutations without repetitions:How many ways can ten people be arranged for a photo lineup in one horizontal line?2.How many different batting orders are there in a 9-person softball team? 3.Lynn has 59 books to arrange on a shelf. How many different ways are there to arrange all the books?Suppose you have n objects to be arranged in a line, how many different permutations (ordered arrangements are possible?4.There are 10 finalists in a figure skating competition. How many ways can a gold, silver, and bronze medal be awarded? Explain in words. Show calculations if necessary.5.The ASB is trying to fill 4 positions for next school year: President, Vice-President, Secretary, and Treasurer. Fifteen students are interested in running for those positions. In how many ways can ASB fill the vacancies? No student can take on more than one position.6.You have 100 pictures in your iPhone. You want to select 24 of them to arrange in your Facebook page. How many different arrangements of pictures are possible?Suppose you are choosing r objects from n objects to be arranged in a line, how many different permutations (ordered arrangements) are possible?7.How many different arrangements of the letters in the word “MATH” are possible? 8.How many different arrangements of the letters in the word “ALGEBRA” are possible?9.How many different arrangements of the letters in the word “MATHEMATICS” are possible?10.How many different arrangements of the letters in the word “STATISTICS” are possible?Suppose n objects are to be arranged in a line where there are r1 indistinguishable objects of style 1, r2 indistinguishable objects of style 2, ... , and rj indistinguishable objects of style j, how many different permutations (ordered arrangements) are possible?Combinations1.Lily is picking two types of fruits for her fruit bowl. She has three choices available: strawberries, grapes, and peaches. How many different fruit combinations can she make?2.Tim has 5 shirts in his closet. He is choosing 3 of them to take on a vacation. How many different ways can he do this?3.In how many ways can four people be selected from a group of six to serve on a committee?4.At Joe's Pizza Parlor, in addition to cheese there are 8 different toppings. a)If you want to order a pizza with only 3 toppings, how many different ways can you order your pizza?b)If you can order any number of those 8 toppings, then how many different combinations of toppings could you possibly order? Suppose you are choosing a group of r objects from n objects, how many different combinations are possible? (Order does not matter. No object is repeated.)5.There are 12 boys and 14 girls in Mrs. Smith’s math class.? a)Find the number of ways that she can select a team of?3 students from the class to work on a group project.? b)Find the number of ways that she can select a team of?2 girls and 2 boys from the class to be on a committee.c)Find the number of ways that she can arrange the students in a class photo.6.Suppose you toss a coin 12 times, recording whether you get heads or tails.a. How many possible sequences of heads and tails are there?b. How many sequences have just one head?c.How many sequences have exactly two heads?d.What is the probability of getting either all heads or all tails in 12 tosses?7.A multiple-choice exam consists of 15 questions, each of which has 4 possible answers with only one right answer. a)If a student guesses on all the questions, what is the probability that the student will have all the questions right?b) If a student guesses on all the questions, what is the probability that the student will have exactly 10 of the questions right?8.In the California Super Lotto Plus lottery, each ticket consists of a set of five numbers chosen from 1-47, in which no number is repeated and the order does not matter; and one MEGA number chosen from 1-27. a)How many different tickets can be formed? You win the jackpot if your ticket has all numbers matching the winning numbers that are randomly drawn by the lottery official. What is your probability of winning the jackpot if you buy only one ticket?b)A ticket matches all five of the first five winning numbers but not the MEGA number. How many different tickets of this kind are possible? What is the probability to get such a ticket?c)You win $1 if your ticket matches three of the first five numbers only but not the MEGA number. How many different tickets of this kind are possible? What is the probability of winning $1 if you only buy one ticket?d)What is the probability that a ticket would match only four of the first five numbers and the MEGA number?10.Without a calculator determine if the following pairs of quantities are equal. If they are not equal, explain which one is bigger. 2100 or 100!100! or 50!50!100! / 50! or 60! / 30!100! / 50! or (100/50)!0! or 1!nP2 or nC2 for all whole numbers n ≥ 2.nP0 or nC0 for all whole numbers n.nCr or nCn-r for all whole numbers n and r where 0 ≤?r ≤?n.nPr or nPn-r for all whole numbers n and r where 0 ≤ r ≤?n. ................
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