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# morecountingkey - McCombs Math 381 More Counting Examples 1...

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McCombs Math 381 More Counting Examples 1. In a technician's box there are 400 VLSI chips, 12 of which are faulty. How many ways are there to pick two chips, so that one is a working chip and the other is faulty? There are 12 1 ! " # \$ % ways to pick a faulty chip. There are 388 1 ! " # \$ % ways to pick a working chip. Total ways = 12 1 ! " # \$ % 388 1 ! " # \$ % = 12 388 = 4656 . 2. How many truth tables are possible for compound propositions with the five variables p , q , r , s , t ? Note that a truth table with five variables must contain 2 5 = 32 rows. There are 2 possible outcomes for each row, i.e. T or F. Thus, there are 2 32 possible five variable truth tables. 3. How many bit strings of length 10 have more 0s than 1s? If there are more 0s than 1s in the bit string, then there are five possible cases. six 0s: 10 6 ! " # \$ % possibilities seven 0s: 10 7 ! " # \$ % possibilities eight 0s: 10 6 ! " # \$ % possibilities nine 0s: 10 6 ! " # \$ % possibilities ten 0s: 10 6 ! " # \$ % possibilities Total = 10 6 ! " # \$ % + 10 7 ! " # \$ % + 10 8 ! " # \$ % + 10 9 ! " # \$ % + 10 10 ! " # \$ % = 1 2 ! " # \$ % 2 10 10 5 ! " # \$ % ! " # \$ % 4. How many subsets with an odd number of elements does a set with 10 elements have? 10 1 ! " # \$ % + 10 3 ! " # \$ % + 10 5 ! " # \$ % + 10 7 ! " # \$ % + 10 9 ! " # \$ % = 2 9 5. How many subsets with more than two elements does a set with 100 elements have? There are a total of 2 100 total subsets. There are 100 0 ! " # \$ % + 100 1 ! " # \$ % + 100 2 ! " # \$ % subsets with cardinality less than or equal to 2. Thus, there are 2 100 ! 100 0 " # \$ % ! 100 1 " # \$ % ! 100 2 " # \$ % subsets with more than two elements.

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McCombs Math 381 More Counting Examples 6. Each user has a password 6 characters long where each character is an uppercase letter, a lowercase letter, or a digit. Each password must contain at least one digit. How long will it take to check every possible character combination, if each check takes one unit of time. There are 26 + 26 + 10 ( ) 6 total password options. There are 26 + 26 ( ) 6 passwords that do not contain a digit. So there are
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morecountingkey - McCombs Math 381 More Counting Examples 1...

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