Ch2-count - Chapter 2 Counting C HAPTER 2 COUNTING COUNTING...

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Chapter 2 Counting 11 HAPTER 2 I. BASIC PRINCIPLES OF COUNTING This chapter is concerned with “counting”; counting the number of elements in a given set or in some specified subset of a given set, counting the possible outcomes of some experiment, or counting the number of ways to perform a sequence of operations. Our objective is to provide systematic approaches to, and methods for answering the question: How many? Examples 1.1: 1. Let } , , { c b a A = . How many subsets of A are there? 2. If we toss a penny and a nickel, how many different possible outcomes are there? 3. An experiment consists of flipping a coin and then rolling a die. How many different possible outcomes are there? Solutions: 1. The subsets of { , , } a b c are: , } { a , } { b , } { c , } , { b a , } , { c a , } , { c b , } , , { c b a . There are 8 subsets. 2. The possible outcomes are HH , HT , TH , TT where, for example, HH signifies “head on the penny” and “ head on the nickel”, etc. There are 4 possible outcomes. 3. The coin comes up in one of two ways; H or T , the die comes up in any one of six ways; 1, 2, 3, 4, 5, 6. The possible outcomes are { H1, H2, … , H6, T1, T2, …, T 6}. There are 2 6 12 ⋅ = possible outcomes. C COUNTING COUNTING COUNTING COUNTING

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Chapter 2 Counting 12 In the examples above, we were able to answer the “how many” question because the number of elements in each set was small and we were able to list the elements explicitly. Here is another, less obvious, example: Example 1.2: In a group of 100 college freshmen, it is found that 50 students are taking English, 30 students are taking mathematics and 10 students are taking both English and mathematics. ( a ) How many students are taking either English or mathematics? That is, how many students are taking at least one of the two courses? ( b ) How many students are taking neither English nor mathematics? Solution: Let E represent the students who are taking English and let M represent the students who are taking mathematics. We might be tempted to answer question ( a ) by 80 30 50 ) ( ) ( ) ( = + = + = M n E n M E n . However, as we now show, this is not correct. Consider the following sequence of Venn diagrams: 1. We know that there are 10 students taking both English and mathematics: 2. Now, since 50 ) ( = E n and 10 of those have already been accounted for, there must be 40 students who are taking English but not mathematics: E M 10 U E M 10 U 40
Chapter 2 Counting 13 3. Similarly, 30 ) ( = M n and 10 have been accounted for in M E , so there must be 20 students who are taking mathematics but not taking English: We can now answer the first question: The number of students who are either taking English or mathematics is given by 70 20 10 40 ) ( = + + = M E n . Note that,

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Ch2-count - Chapter 2 Counting C HAPTER 2 COUNTING COUNTING...

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