l05_usrepl

l05_usrepl - Sampling III: Unordered Sampling with...

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02/02/2009 CS206 - Intro. to Discrete Structures II 1 Sampling III: Unordered Sampling with Replacement Reading: Ross, Ch 1., Sec. 5. Rosen, Ch 5., Sec. 5. Monday, February 2, 2009
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02/02/2009 CS206 - Intro. to Discrete Structures II 2 Example 1 • How many ways to pick four pieces of fruit from a bowl of apples, oranges, and pears? Order does not matter, only the type of fruit. (There enough pieces to pick four of each type.) Do we need to know how many pieces of fruit in the bowl? Say 50. Is the answer C(50,4)? No! Because not all 50 are different. Monday, February 2, 2009
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02/02/2009 CS206 - Intro. to Discrete Structures II 3 Example 1 (cont’d) • Consider this: There are three bins, one each for A, O, and P. Need to distribute four (generic) pieces of fruit into the three bins. Fruit (generic) A O P A O P A O P Monday, February 2, 2009
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02/02/2009 CS206 - Intro. to Discrete Structures II 4 Example 1 (cont’d) • Or consider this… Place two bars between four pieces of fruit. (4+2)! / [ 4! ¢ 2! ] = 6! / [ 4! ¢ 2! ] = 15 = C(6,4) - pick 4 pieces from a set of 4+3-1 A O P , Monday, February 2, 2009
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CS206 - Intro. to Discrete Structures II 5 Combinations with Repetition • Given a set of r types of elements, there are C(n+r-1,r-1) ways to choose n elements, with repetition. E.g., Fruit example: r = 3 types of fruit. Ways of picking n = 4 pieces:
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l05_usrepl - Sampling III: Unordered Sampling with...

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