# lec05 - AMS 301 Lecture 5 AMS Ning SUN Jul 30 2009 Two...

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AMS 301 Lecture 5 Ning SUN Jul 30 2009

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Two Basic Counting Principles
Warmup There are 10 creamery flavors at Cold Stone, how many different creations can you make? 3 AMS301, Summer 2009, Ning SUN

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The Addition Principle r 1 different objects in the first set, r 2 different objects in the second set, …, r m different objects in the m th set, # of ways to select an object form one of the m sets: r 1 + r 2 +…+ r m . disjoint 4 AMS301, Summer 2009, Ning SUN
Outcomes Outcomes ->“disjoint pieces” >“disjoint pieces” No overlap + complete Enumeration 5 AMS301, Summer 2009, Ning SUN

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Example You want to take a course from the list: 6 AMS courses 4 MAT courses 2 CHE courses 2 BIO courses # of ways to choose a course: 6 + 4 + 2 + 2 = 14. 6 AMS301, Summer 2009, Ning SUN
The Multiplication Principle A procedure -> m successive (ordered) stages: r 1 different outcomes in the first stage, r 2 different outcomes in the second stage, …, r m different outcomes in the m th stage. # of different composite outcomes the total procedure: r 1 r 2 r m . Independent distinct 7 AMS301, Summer 2009, Ning SUN

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Procedure Procedure ->“sequential stages” >“sequential stages” Independent Distinct 8 AMS301, Summer 2009, Ning SUN
Example You have: 6 different ties 4 different shirts 2 different pants 2 different shoes # of combinations: 6 4 2 2 = 96. 9 AMS301, Summer 2009, Ning SUN

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Comments on the principles The counting principles are simple, but powerful and easy to misuse. A lot of times we need to combine the two principles to solve problems. 1 0 AMS301, Summer 2009, Ning SUN
Example: Rolling Dice Q: Two six-sided dice are rolled, one red and one white. How many different outcomes are there? Each die has six outcomes. The outcomes are independent. Multiplication Principle: 6 6 = 36 Note: “ Independent ” means that what we roll on the first die does not influence what we roll on the second. 1 1 AMS301, Summer 2009, Ning SUN

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Example: Rolling Dice (cont’d) Q: What is the probability that there are no doubles? # total outcomes = 36 # desired outcomes? stage 1: 6 stage 2: 5 Probability = 30 / 36 = 5/6 Probability = # desired outcomes # total outcomes Multiplication : 6 5 = 30 1 2 AMS301, Summer 2009, Ning SUN
Example: Example: Arranging Books Arranging Books There are 5 different Spanish books, 6 different French books, 8 different Transylvanian books. Q: How many different ways are there to pick an (unordered) pair of two books not in the same language? S F T F S T 1 3 AMS301, Summer 2009, Ning SUN

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Example: Example: Arranging Books (cont’d) Arranging Books (cont’d) 3 types of combinations 5 6 = 30 5 8 = 40 6 8 = 48 These 3 types are disjoint. # ways to pick a pair: 30 + 40 + 48 = 118 5S 6F 8T 6F 5S 8T Multiplication Addition 1 4 AMS301, Summer 2009, Ning SUN
Example: Example: Sequences of Letters Sequences of Letters Q1: How many ways are there to obtain a three- letter sequence using the letters a, b, c, d, e, f with repetition of letters allowed ?

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## This note was uploaded on 02/13/2011 for the course AMS 301 taught by Professor Arkin during the Spring '08 term at SUNY Stony Brook.

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lec05 - AMS 301 Lecture 5 AMS Ning SUN Jul 30 2009 Two...

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