Counting Techniques. Multiplication Principle (also called the Fundamental Counting Principle) Combinations Permutations Number of subsets of a given.

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Presentation transcript:

Counting Techniques

Multiplication Principle (also called the Fundamental Counting Principle) Combinations Permutations Number of subsets of a given set

Suppose there are n different decisions and each decision has choices where i is some number from 1 to n. Then the overall total number of ways in which those n decisions can be made is the product Multiplication Principle

Examples Suppose you have 7 different shirts, 5 different pairs of pants and 3 pairs of shoes. How many outfits are possible? Suppose there are 10 questions on a multiple-choice exam and each question can be answered in 5 different ways (A, B, C, D or E). How many ways are there to complete the exam assuming every question is answered?

Examples Suppose you have 7 different shirts, 5 different pairs of pants and 3 pairs of shoes. How many outfits are possible? –Answer: 7*5*3=105 possible outfits Suppose there are 10 questions on a multiple-choice exam and each question can be answered in 5 different ways (A, B, C, D or E). How many ways are there to complete the exam assuming every question is answered? –Answer: ways to complete the exam.

Combinations The number of ways of choosing r distinct objects from n distinct objects is given by the formula Note and 0! = 1

Examples How many ways can 3 movies be chosen from a list of 5 movies? A committee consists of 10 people. How many ways are there to form a coalition of 5 people from the committee?

Examples How many ways can 3 movies be chosen from a list of 5 movies? –Answer: A committee consists of 10 people. How many ways are there to form a coalition of 5 people from the committee? –Answer:

Permutations The number of ways of selecting r distinct objects from n distinct objects and rearranging those r objects is given by the formula

Examples Suppose there are 10 movies playing in the theater. How many ways are there of selecting and ranking your favorite 3? There are 5 people in a coalition of voters. How many ways are there to rearrange those 5 people in distinct orderings?

Examples Suppose there are 10 movies playing in the theater. How many ways are there of selecting and ranking your favorite 3? –Answer: There are 5 people in a coalition of voters. How many ways are there to rearrange those 5 people in distinct orderings? –Answer: 5! = 120 ways

Given a set with n elements, the number of subsets of the given set is. Examples: –Let A = {x, y, z}. How many subsets does A have? –Suppose a committee consists of 3 people. How many possible coalitions can be formed from this committee? Number of Subsets

Examples: –Let A = {x, y, z}. How many subsets does A have? Answer: subsets –Suppose a committee consists of 3 people. How many possible coalitions can be formed from this committee? Answer: coalitions Number of Subsets