Counting Techniques. Fundamental Counting Principal Two Events: If one event can occur in m ways and another event can occur in n ways, then the number.

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Presentation transcript:

Counting Techniques

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)

Tree Diagrams Look at Page 701 of text book

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.

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.

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?

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

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?

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?

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:

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?

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:

Example 4: Finding Permutations with Repetition Find the number of distinguishable permutations of the letters in A) ALGEBRA and B)MATHEMATICS.