lec4r - Two Basic Counting Principles Summation Principle:...

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Two Basic Counting Principles Summation Principle : If a complex action can be performed using one of k alternative methods, m 1 , . . . , m k , and the methods can be performed in n 1 , . . . , n k ways, respectively, then the complex action can be peformed in n 1 + . . . + n k ways. Multiplication Principle : If a complex action can be broken down into a series of k component actions, peformed one after the other, and the first can be performed in n 1 ways, the second in n 2 ways, . . . , and the last in n k ways, then the complex action can be peformed in n 1 n 2 ··· n k ways. 1
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Combinations: Examples Example 1 How many ways can you draw a heat or a diamond from a standard deck of cards? A: 13+13=26 Example 2 If two dies are thrown one after the other, how many ways can you get sum of 4 or a sum of 8? A: 3+5=8 Example 3 How many different license plates are possible if each contains a sequence of 3 letters followed by 3 digits (if no sequence of letters or numbers are prohibited)? A: 26 × 26 × 26 × 10 × 10 × 10 = 17576000 2
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Permutations and Combinations Ordered Samples With Replacement Experiment: A box has n items numbered 1 , . . . , n . Draw k items with replacement. (A number can be drawn twice). Sample Space: Ω = { ( x 1 , . . . , x k ) : x i ∈ { 1 , . . . , n }} = { x 1 x 2 ...x k : x i ∈ { 1 , . . . , n }} What is | | ? Break the complex action into a series k single draws. ( x i is outcome on draw i ). Then n possibilities for x 1 , n possibilities for x 2 , . . . , n possibilities for x k . Then, multiplication principle gives us | | = n · n ··· n = n k . 3
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Ordered Samples With Replacement: Examples Example 1: Octal Numbers A five-digit octal number is a 5-digit number consisting of the digits 0 , . . . , 7 .
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lec4r - Two Basic Counting Principles Summation Principle:...

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