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Algebra 2: Section 12.1 The Fundamental Counting Principle and Permutations

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Examples for types of problems Experiment toss a coin roll a die toss two coins draw a card select a day of the week Experiment toss a coin roll a die toss two coins draw a card select a day of the week Sample Space o head or tail o 1, 2, 3, 4, 5, or 6 o HH, HT, TH, or TT o 52 possible outcomes o Sun, Mon, Tues, Wed, Thurs, Fri, or Sat

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Example You have 12 shirts, 7 pairs of pants, and 3 pairs of shoes. How many different combinations of outfits can you possibly have?

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Fundamental Counting Principle If one event occurs in m ways and another occurs in n ways, then the number of ways that both events can occur is mn. This also extends to more than two events Ex: mnp for three events that occur in m, n, and p ways If one event occurs in m ways and another occurs in n ways, then the number of ways that both events can occur is mn. This also extends to more than two events Ex: mnp for three events that occur in m, n, and p ways

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The Fundamental Counting Rule How many possible ZIP codes are there? = 100,000 How many possible ZIP codes are there if 0 can not be used for the first digit? = 90,000 How many possible ZIP codes are there? = 100,000 How many possible ZIP codes are there if 0 can not be used for the first digit? = 90,000

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Example If there are five students who can receive five different scholarships, how many different ways can the five scholarships be awarded? How many different ways can the scholarships be awarded if each student can only win one scholarship? If there are five students who can receive five different scholarships, how many different ways can the five scholarships be awarded? How many different ways can the scholarships be awarded if each student can only win one scholarship?

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Factorial Definition n! = n(n-1)(n-2)…321 Examples: 4! 10! 1! = ____ 0! = ____ n! = n(n-1)(n-2)…321 Examples: 4! 10! 1! = ____ 0! = ____ = 4321= 24 = 3,628,800=

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Example The standard configuration for a New York license plate is 3 digits followed by 3 letters. How many different license plates are possible if digits and letters can be repeated? How many different license plates are possible if digits and letters cannot be repeated? The standard configuration for a New York license plate is 3 digits followed by 3 letters. How many different license plates are possible if digits and letters can be repeated? How many different license plates are possible if digits and letters cannot be repeated?

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PermutationsPermutations An ordering of objects Placing a certain number of objects in a certain number of positions Notation: n P r nr (n things placed in r positions) Example: How many ways can you place 5 different students in 3 different chairs? An ordering of objects Placing a certain number of objects in a certain number of positions Notation: n P r nr (n things placed in r positions) Example: How many ways can you place 5 different students in 3 different chairs?

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Permutation Example There are 8 different people running in the finals of the 400M dash. How many different ways can they place 1st, 2nd, and 3rd?

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The Permutation Rule Example Ex: In how many ways can a batting order be made from a team of 17 softball players?

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The Fundamental Counting Rule Extra Example: How many possible ways can three cards be drawn from a deck, without replacing them? = 132,600 Extra Example: How many possible ways can three cards be drawn from a deck, without replacing them? = 132,600

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Permutations Examples Extra Example: Six runners are in the first heat of the 100 meter sprint. In how many different ways can the race end? = 720 ways In how many ways can the first three spots of the race be filled? 654 = 120 ways Extra Example: Six runners are in the first heat of the 100 meter sprint. In how many different ways can the race end? = 720 ways In how many ways can the first three spots of the race be filled? 654 = 120 ways

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Fundamental Counting Principle If one event occurs in m ways and another occurs in n ways, then the number of ways that both events can occur is mn. This also extends to more than two events Ex: mnp for three events that occur in m, n, and p ways If one event occurs in m ways and another occurs in n ways, then the number of ways that both events can occur is mn. This also extends to more than two events Ex: mnp for three events that occur in m, n, and p ways

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Factorial Definition n! = n(n-1)(n-2)…321 Examples: 4! 10! n! = n(n-1)(n-2)…321 Examples: 4! 10! = 4321= 24 = 3,628,800=

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PermutationsPermutations An ordering of objects Placing a certain number of objects in a certain number of positions Notation: n P r nr (n things placed in r positions) Example: How many ways can you place 5 different students in 3 different chairs? An ordering of objects Placing a certain number of objects in a certain number of positions Notation: n P r nr (n things placed in r positions) Example: How many ways can you place 5 different students in 3 different chairs?

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Example How many distinguishable permutations of the letters in MATH are there? MATH How many distinguishable permutations of the letters in MATH are there? MATH

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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 is:

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Permutations with Repetition Example How many distinguishable permutations of the letters in BASKETBALL are there?

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Permutations with Repetition Example How many distinguishable permutations of the letters in MISSISSIPPI are there?

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