a2-21-08

a2-21-08 - Answers to Homework 21 February 2008 Section 17...

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21 February 2008 Section 17 1. Evaluate 3 2 ±± and 2 3 ±± by explicitly listing all possible multisets of the appropriate size. Check that your answers agree with the formula in The- orem 17.8. For 3 2 ±± the multisets are h 1 ; 1 i , h 2 ; 2 i , h 3 ; 3 i , h 1 ; 2 i , h 1 ; 3 i , h 2 ; 3 i There are six of them, and the formula in Theorem 17.8 gives ²² 3 2 ³³ = ² 4 2 ³ = 6 For 2 3 ±± the multisets are h 1 ; 1 ; 1 i , h 1 ; 1 ; 2 i , h 1 ; 2 ; 2 i , h 2 ; 2 ; 2 i There are four of them, and the formula in Theorem 17.8 ²² 2 3 ³³ = ² 4 3 ³ = 4 2. Give the stars-and-bars representation for all the sets you found in the previous problem . h 1 ; 1 i h 2 ; 2 i h 3 ; 3 i h 1 ; 2 i h 1 ; 3 i h 2 ; 3 i h 1 ; 1 ; 1 i h 1 ; 1 ; 2 i h 1 ; 2 ; 2 i h 2 ; 2 ; 2 i 4. What multiset is encoded by the stars-and-bars notation ? There are four stars, so we are choosing a multiset of size 4 . There are three bars, so we are choosing from a set of size 4 , which we might as well take to be f 1 ; 2 ; 3 ; 4 g . So the indicated multiset is h 1 ; 4 ; 4 ; 4 i . There is one
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a2-21-08 - Answers to Homework 21 February 2008 Section 17...

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