basic_counting_principleskey - McCombs Math 381 Basic...

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McCombs Math 381 Basic Counting Principles 1 1. The Product Rule: Suppose a procedure can be broken down into a finite sequence of tasks, T 1 , T 2 , T 3 , ! ! ! , T m , where each task in the sequence can be performed in a finite number of ways, regardless of how the previous tasks were done. In particular, suppose T 1 can be performed in n 1 ways, T 2 can be performed in n 2 ways, T 3 can be performed in n 3 ways, . .., T m can be performed in n m ways. Given these criteria, there are a total of n 1 ! n 2 ! n 3 ! ! ! ! n m different ways to perform the procedure. 2. The Sum Rule: Suppose the finite set T = the set containing all possible ways to perform a given task. Suppose also that T = A 1 ! A 2 ! A 3 !"""! A m , where A 1 , A 2 , A 3 , ! ! ! , A m is a finite sequence of finite, disjoint sets. Given these criteria, there are a total of A 1 + A 2 + A 3 + ! ! ! ! + A m different ways to perform task. 3. A permutation is an ordered sequence of elements chosen from a set. 4. A permutation of length r from a set with n elements is called an r -permutation . The number of different r -permutations from a set with n elements is denoted by P n , r ( ) . Important Theorem: Given positive integers n and r , with 1 ! r ! n , P n , r ( ) = n r , if repetition is allowed n ! n ! r ( ) ! , if repetition is not allowed " # $ % $ Note that n ! n ! r ( ) ! = n n ! 1 ( ) n ! 2 ( ) """ n ! r + 1 ( ) . 5. A subset (or combination ) is an unordered selection of elements chosen from a set. 6. A set containing n elements will yield a total of 2 n different subsets. 7. The number of different r -combinations , chosen from a set of n elements is denoted by C n , r ( ) . Important Theorem: Given positive integers n and r, with 1 ! r ! n , C n , r ( ) = n r ! " # $ % = n ! r ! n r ( ) ! 8. Inclusion-Exclusion: Given finite sets A and B , A B A B A B ! = + " # .
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McCombs Math 381 Basic Counting Principles 2 Examples 1. How many 8-digit numbers are there in base 10 that do not have two consecutive digits the same? (Note: Leading zeros are not permitted.) We are constructing a list of length 8 from the set {0,1,2,3,4,5,6,7,8,9}. Since the leading digit cannot be 0, there are 9 choices for that digit. 2 nd digit: 9 choices, since we cannot repeat the previous digit. 3 rd digit: 9 choices, since we cannot repeat the previous digit. 4 th digit: 9 choices, since we cannot repeat the previous digit. . . . . . . 8 th digit: 9 choices, since we cannot repeat the previous digit. So we have 8 9 43,046,721 = possibilities. 2. How many 5-digit numbers are there in base 3 arithmetic? (Note: Leading zeros are permitted.) We are constructing a list of length 5 from the set {0,1,2}.
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basic_counting_principleskey - McCombs Math 381 Basic...

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