Problem Set 4 Solutions

# Problem Set 4 Solutions - CSE 21: Solutions - Problem Set 4...

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Unformatted text preview: CSE 21: Solutions - Problem Set 4 1. These problems are variations of Bars and Stars . Let x 1 + x 2 + x 3 + x 4 = n be the equation. (a) This is of the standard form: there are n stars, and we need 3 bars. The answer is ( n +3 3 ) . (b) If all four variable are positive, transform the equation to ( x 1- 1)+( x 2- 1)+( x 3- 1)+( x 4- 1) = n- 4. Change variables: x i- 1 y i . Then y 1 + y 2 + y 3 + y 4 = n- 4. There is a bijection between positive solutions to x i s and non-negative solutions to y i s. Thus the number of solutions (to both equatiosn) is ( n- 4+3 3 ) . The probability is ( n- 1 3 ) / ( n +3 3 ) . (c) Transform the equation to ( x 1 +2)+( x 2 +2)+( x 3 +2)+( x 4 +2) = n +8. The answer is ( n +8+3 3 ) . (d) Transform the equation to (8- x 1 ) + (8- x 2 ) + (8- x 3 ) + (8- x 4 ) = 32- n . The answer is ( 32- n +3 3 ) . 2. (a) Let m be the size of the multisets, and n be the alphabet size. As shown in class, we can either use Stars and Bars , or transform it into counting subsets. The answer is, or transform it into counting subsets....
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## This note was uploaded on 12/03/2009 for the course CSE 21 taught by Professor Graham during the Spring '07 term at UCSD.

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Problem Set 4 Solutions - CSE 21: Solutions - Problem Set 4...

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