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Chapter 8 Counting Principles: Further Probability Topics Section 8.1 The Multiplication Principle; Permutations

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Warm – Up for Sections 8.1 and 8.2 A certain game at an amusement card consists of a person spinning a spinner, choosing a card, and then tossing an unbiased coin. Prizes are awarded based on the combination created from performing each of the three tasks. The spinner has three equal areas represented by Purple, Gold, and Red; the cards to choose from include a King, Queen, and Joker; and the coin has a Crown on one side and a Donkey on the other. How many possible outcomes are there? If the order in which the tasks were performed made a difference, would there be more outcomes or fewer outcomes?

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Alice can’t decide what to wear between a pair of shorts, a pair of pants, and a skirt. She has four tops that will go with all three pieces: one red, one black, one white, and one striped. Alice can’t decide what to wear between a pair of shorts, a pair of pants, and a skirt. She has four tops that will go with all three pieces: one red, one black, one white, and one striped. How many different outfits could Alice create from these items of clothing?

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BottomTopOutfit Shorts Pants Skirt Red Black White Striped Red Black White Striped Red Black White Striped Shorts, Red Top Shorts, Black Top Shorts, White Top Shorts, Striped Top If the tree diagram is finished, how many outfits will she have? 12 outfits!!

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Tree diagrams are not often convenient, or practical, to use when determining the number of outcomes that are possible. Tree diagrams are not often convenient, or practical, to use when determining the number of outcomes that are possible. Rather than using a tree diagram to find the number of outfits that Alice had to choose from, we could have used a general principle of counting: the multiplication principle. Rather than using a tree diagram to find the number of outfits that Alice had to choose from, we could have used a general principle of counting: the multiplication principle.

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Multiplication Principle

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Alice can’t decide what to wear between a pair of shorts, a pair of pants, and a skirt. She has four tops that will go with all three pieces: one red, one black, one white, and one striped. Alice can’t decide what to wear between a pair of shorts, a pair of pants, and a skirt. She has four tops that will go with all three pieces: one red, one black, one white, and one striped. How many different outfits could Alice create from these items of clothing? Using the multiplication principle, we multiply the number of options she has for what to wear on “bottom” and the number of options she has for what to wear on “top”. 3 bottoms 4 tops = 12 outfits

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A product can be shipped by four airlines and each airline can ship via three different routes. How many distinct ways exist to ship the product? A product can be shipped by four airlines and each airline can ship via three different routes. How many distinct ways exist to ship the product?

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How many different license plates can be made if each license plate is to consist of three letters followed by three digits and replacement is allowed? How many different license plates can be made if each license plate is to consist of three letters followed by three digits and replacement is allowed? ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ L L L D D D L L L D D D If replacement is not allowed? If replacement is not allowed? 26 10 = 26 ³ 10 ³ = 17, 576, 000 ___ ___ ___ ___ ___ ___ L L L D D D L L L D D D 26 25 24 10 9 8 = 11, 232, 000

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How many different license plates can be made if each license plate begins with 63 followed by three letters and two digits? How many different social security numbers are possible if the first digit may not be zero?

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Marie is planning her schedule for next semester. She must take the following five courses: English, history, geology, psychology, and mathematics. a.) In how many different ways can Marie arrange her schedule of courses? her schedule of courses? b.) How many of these schedules have mathematics listed first? listed first?

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You are given the set of digits {1, 3, 4, 5, 6}. a.) How many three-digit numbers can be formed? b.) How many three-digits numbers can be formed if the number must be even? the number must be even? c.) How many three-digits numbers can be formed if the number must be even and no repetition of the number must be even and no repetition of digits is allowed? digits is allowed?

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A certain Math 110 teacher has individual photos of each of her three dogs: Indy, Sam, and Jake. In how many ways can she arrange these photos in a row on her desk?

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Factorial Notation

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If seven people board an airplane and there are nine aisle seats, in how many ways can the people be seated if they all choose aisle seats?

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Permutations A permutation of r (where r ≥ 1) elements from a set of n elements is any specific ordering or arrangement, without repetition, of the r elements. A permutation of r (where r ≥ 1) elements from a set of n elements is any specific ordering or arrangement, without repetition, of the r elements. Each rearrangement of the r elements is a different permutation. Each rearrangement of the r elements is a different permutation. Permutations are denoted by nPr or P(n, r) Permutations are denoted by nPr or P(n, r) Clue words: arrangement, schedule, order, awards, officers Clue words: arrangement, schedule, order, awards, officers

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A disc jockey can play eight records in a 30- minute segment of her show. For a particular 30-minute segment, she has 12 records to select from. In how many ways can she arrange her program for the particular segment? A disc jockey can play eight records in a 30- minute segment of her show. For a particular 30-minute segment, she has 12 records to select from. In how many ways can she arrange her program for the particular segment? A chairperson and vice-chairperson are to be selected from a group of nine eligible people. In how many ways can this be done? A chairperson and vice-chairperson are to be selected from a group of nine eligible people. In how many ways can this be done?

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Distinguishable Permutations If the n objects in a permutation are not all distinguishable – that is, if there so many of type 1, so many of type 2, and so on for r different types, then the number of distinguishable permutations is If the n objects in a permutation are not all distinguishable – that is, if there so many of type 1, so many of type 2, and so on for r different types, then the number of distinguishable permutations is n!. n!. n ! n ! n ! n ! n ! n ! 1 r 2

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How many distinct arrangements can be formed from all the letters of SHELTONSTATE? Step 1: Count the number of letters in the word, including repeats. 12 letters Step 2: Count the number of repetitious letters and the number of times each letter repeats. S : 2 repeats E : 2 repeats T : 3 repeats Solution: 12!. 2! 2! 3! = 19, 958, 000

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In how many distinct ways can the letters of MATHEMATICS be arranged? In how many distinct ways can the letters of MATHEMATICS be arranged? In how many distinct ways can the letters of BUCCANEERS be arranged? In how many distinct ways can the letters of BUCCANEERS be arranged?

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