3 4 Counting Principles Fundamental Counting Principle If

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§ 3. 4 Counting Principles

§ 3. 4 Counting Principles

Fundamental Counting Principle If one event can occur in m ways and a second

Fundamental Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m· n. This rule can be extended for any number of events occurring in a sequence. Example: A meal consists of a main dish, a side dish, and a dessert. How many different meals can be selected if there are 4 main dishes, 2 side dishes and 5 desserts available? # of main # of side dishes 4 2 There are 40 meals available. # of desserts 5 Larson & Farber, Elementary Statistics: Picturing the World, 3 e = 40 2

Fundamental Counting Principle Example: Two coins are flipped. How many different outcomes are there?

Fundamental Counting Principle Example: Two coins are flipped. How many different outcomes are there? List the sample space. Start 1 st Coin Tossed Heads 2 ways to flip the coin Tails 2 nd Coin Tossed Heads Tails 2 ways to flip the coin There are 2 2 = 4 different outcomes: {HH, HT, TH, TT}. Larson & Farber, Elementary Statistics: Picturing the World, 3 e 3

Fundamental Counting Principle Example: The access code to a house's security system consists of

Fundamental Counting Principle Example: The access code to a house's security system consists of 5 digits. Each digit can be 0 through 9. How many different codes are available if a. ) each digit can be repeated? b. ) each digit can only be used once and not repeated? a. ) Because each digit can be repeated, there are 10 choices for each of the 5 digits. 10 · 10 = 100, 000 codes b. ) Because each digit cannot be repeated, there are 10 choices for the first digit, 9 choices left for the second digit, 8 for the third, 7 for the fourth and 6 for the fifth. 10 · 9 · 8 · 7 · 6 = 30, 240 codes Larson & Farber, Elementary Statistics: Picturing the World, 3 e 4

Terms Permutation – an ordered arrangement of objects (e. g. your cell phone pass

Terms Permutation – an ordered arrangement of objects (e. g. your cell phone pass code) Combination – a selection of r objects from a group of n things when order does not matter (e. g. a combo meal from a fast food restaurant) Factorial – the product of an integer, and ALL the positive integers before it (countdown multiplication). An exclamation mark is used as the operation sign. (e. g. 4! = 4 x 3 x 2 x 1 = 24) Larson & Farber, Elementary Statistics: Picturing the World, 3 e 5

Permutations A permutation is an ordered arrangement of objects. The number of different permutations

Permutations A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n!. “n factorial” Example: How many different surveys are required to cover all possible question arrangements if there are 7 questions in a survey? 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040 surveys Larson & Farber, Elementary Statistics: Picturing the World, 3 e 6

Checkpoint 1. The starting lineup for a baseball team consists of nine players. How

Checkpoint 1. The starting lineup for a baseball team consists of nine players. How many different batting orders are possible using the starting lineup? 2. The teams in the National League Central Division are: Chicago Cubs, Cincinnati Reds, Houston Astros, Milwaukee Brewers, Pittsburgh Pirates, and St. Louis Cardinals. How many different final standings are possible? Larson & Farber, Elementary Statistics: Picturing the World, 3 e 7

Permutation of n Objects Taken r at a Time The number of permutations of

Permutation of n Objects Taken r at a Time The number of permutations of n elements taken r at a time is # in the group # taken from the group Example: You are required to read 5 books from a list of 8. In how different orders can you do so? Larson & Farber, Elementary Statistics: Picturing the World, 3 e many 8

Checkpoint 1. Find the number of ways of forming 3 -digit codes in which

Checkpoint 1. Find the number of ways of forming 3 -digit codes in which no digit is repeated. 2. In a race with 8 horses, how many ways can 3 horses finish in 1 st, 2 nd, or 3 rd place? (Assume no ties) 3. Forty-three race cars started the 2004 Daytona 500. How many ways can the cars finish 1 st, 2 nd, and 3 rd? (Assume no ties) Larson & Farber, Elementary Statistics: Picturing the World, 3 e 9

Combination of n Objects Taken r at a Time A combination is a selection

Combination of n Objects Taken r at a Time A combination is a selection of r objects from a group of n things when order does not matter. The number of combinations of r objects selected from a group of n objects is # in the collection # taken from the collection Example: You are required to read 5 books from a list of 8. In how many different ways can you do so if the order doesn’t matter? Larson & Farber, Elementary Statistics: Picturing the World, 3 e 12

Application of Counting Principles Example: In a state lottery, you must correctly select 6

Application of Counting Principles Example: In a state lottery, you must correctly select 6 numbers (in any order) out of 44 to win the grand prize. a. ) How many ways can 6 numbers be chosen from the 44 numbers? b. ) If you purchase one lottery ticket, what is the probability of winning the top prize? a. ) b. ) There is only one winning ticket, therefore, Larson & Farber, Elementary Statistics: Picturing the World, 3 e 13

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