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Counting Techniques
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Fundamental Counting Principal Two Events: If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is m·n. Three or More Events: the Fundamental Counting Principal can be extended to three or more events (m·n·p)
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Tree Diagrams Look at Page 701 of text book
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Example 1: Counting Techniques A) You are eating at a banquet. Your choices are 3 different entreés: chicken, fish, vegetarian and 2 sides: salad or mashed potatoes. Use a tree diagram to find the number of choices you have for your meal.
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Example 1: continued B) A high school has 273 freshmen, 291 sophomores, 252 juniors, and 237 seniors. Use the Fundamental Counting Principle to find how many different ways a committee of 1 freshmen, 1 sophomore, 1 junior and 1 senior can be chose.
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Example 1: continued C) A town has telephone numbers that begin with 432 or 437 followed by four digits. How many different telephone numbers are possible if the last four digits cannot be repeated?
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Permutations An ordering of n objects. The Fundamental Counting Principle can be used to determine the number of permutations of n objects. In general, the number of permutations of n distinct objects is N! = n·(n-1)·(n-2)·…·3·2·1
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Example 2: Finding the Number of Permutations A) You have homework assignments from 5 different classes to complete this weekend. In how many different ways can you complete the assignments?
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Example 2: continued B) Twenty-six golfers are competing in a final round of a golf competition. How many different ways can 3 of the golfers finish first, second, and third?
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Permutations of n objects taken r at a time The number of permutations of r objects taken from a group of n distinct objects is denoted by and is given by:
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Example 3: Finding Permutations of n Objects taken r at a Time There are 9 players on a baseball team. In how many ways can you chose A) the batting order of all 9 players and B) a pitcher, catcher, and shortstop from the 9 players?
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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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Example 4: Finding Permutations with Repetition Find the number of distinguishable permutations of the letters in A) ALGEBRA and B)MATHEMATICS.
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