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Chris Morgan, MATH G160 January 27, 2012 Lecture 8 Chapter 4.1: Permutations 1

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We use permutations when we are interested in the number of possible ways to order something and ORDER IS IMPORTANT! When order is not important, then it is a combination n – total number of objects to choose from r – number of times you choose an object Thus a permutation is an ordered arrangement of r objects from a group of m objects. 2

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Permutation example Suppose I have 8 different colors of gumballs. How many different ways can I give 3 children a gumball? 8 * 7 * 6 = 336 3

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Permutation example Does this look like we simply used the BCR? - BCR and Permutation are related. - Lets find out why using the permutation formula: 4

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Sampling Replacement The outcome of a permutation depends on two things. Do we sample: With replacement? or Without replacement? 5

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Permutations with Replacement These arent popular, but lets see what one might look like: How many possible ways could I select from the letters PURDUE if I sample with replacement? _ _ _ 6*6*6*6*6*6 = 6 6 =

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Permutations without Replacement How many different letter arrangements can be formed using the letters BOILERS? Theorem: Special Permutations Rule is the number of ways to order n distinct objects _ _ _ _ _ _ _ So, there are 7! = 5040 ways to arrange the letters BOILERS. 7

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Theorem There are: different permutations of m objects of which m 1, m 2, … m k are alike respectively. 8

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A little bit harder now How many different ways to reorder the letters in the word Statistics? _ _ _ _ _ How many different ways to reorder the letters in the word Probability? 9

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Example (XVIII) At an academic conference, 12 faculties are going to take a picture together. There are 3 professors, 5 associate professors and 4 assistant professors. If we want people at the same level to stay together, how many ways to line them up? 10

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Example (CLII) My new bike lock has three dials numbered between 0 and 9. How many different ways can the code be set if: No restrictions at all? None of the numbers may be the same? No two consecutive numbers may be the same? The third number must be lower than the second? 11

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Example (XLII) You are required to select a 6-character case-sensitive password for an online account. Each character could be upper-case or lower-case letter or a number from 0 to 9. No restrictions at all? The first character can not be a number? The last four characters must all be different? There must be at least one capital letter and at least one number? 12

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Example (MD) I want to have four friends over (five including myself) and I want to make sure none of us bring the same type of liquor. The VBS next to me sells 17 types of liquor. Whats the probability no two people bring the same kind of liquor? 13

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