tucker 5-4 - Applied Combinatorics, 4rth Ed. Alan Tucker...

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4/5/05 Tucker, Sec. 4.3 1 Applied Combinatorics, 4rth Ed. Alan Tucker Section 5.4 Distributions Prepared by Jo Ellis-Monaghan
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4/5/05 Tucker, Sec. 4.3 2 Distributions A distribution problem is an arrangement or selection problem with repetition. Specialized distribution problems must be broken up into subcases that can be counted in terms of simple permutations and combinations (with and without repetition). General guidelines for modeling distributions: Distributions of distinct objects are equivalent to arrangements Distributions of identical objects are equivalent to selections.
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4/5/05 Tucker, Sec. 4.3 3 Basic Models for Distributions Distinct Objects: The process of distributing r distinct objects into n different boxes is equivalent to putting the distinct objects in a row and then stamping one of the n different box names on each object. Thus there are n * n *…* n = n r distributions of the r distinct objects. r  distinct objects different boxes     Red    Red    Blue   Green
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4/5/05 Tucker, Sec. 4.3 4 Specified amount in each box If r i objects must go in box i , then there are P( r ; r 1 , r 2 , …, r n ) distributions. 6 distinct objects 3 1 2 How many in each box ( 29 6 6 3 6 3 1 6;3,1,2 3 1 2 P - - -    =      
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4/5/05 Tucker, Sec. 4.3 5 Basic Models for Distributions Identical Objects: The process of distributing r identical objects into n different boxes is equivalent to choosing an (unordered) subset of r box names with repetition from among the n choices of boxes. Thus there are C ( r + n -1, r ) = ( r + n -1)!/ r !( n -1)! distributions of the r identical objects. Red  Red    Blue  Blue    Green  Green  Green identical  objects
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4/5/05 Tucker, Sec. 4.3 6 Equivalent Forms for Selection with Repetition 1. The number of ways to select r objects with repetition from n different types of objects. 2. The number of ways to distribute r identical objects into n distinct boxes. 3. The number of nonnegative integer solutions to x 1 + x 2 +… + x n = r . Red  Red   
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This note was uploaded on 05/04/2011 for the course MATH 352 taught by Professor Zhang during the Spring '11 term at Saint Mary-of-the-Woods College.

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tucker 5-4 - Applied Combinatorics, 4rth Ed. Alan Tucker...

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