CSE-21.ProblemSet_4-1

CSE-21.ProblemSet_4-1 - Jonathan Thomas Guidry CSE-21 1. (4...

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Jonathan Thomas Guidry CSE-21 1. (4 pts) The equation x 1 + x 2 + x 3 + x 4 = 20 has four distin- guishable variables. (a) How many non-negative integer solutions are there? (b) Let the set of all the non-nagative integer solutions be the sample space. What is the probability that all four variables in a solution are positive? (c) How many integer solutions are there in which no vari- able is less than - 2? (d) How many integer solutions are there in which no vari- able is greater than 8? Hint: These problems are variations of Bars and Stars. 2. (3 pts) Count each of the following (a) the number of multisets of size 10 whose elements lie in { a , b , c , d } ; (b) the number of weakly increasing functions from [ 8 ] to [ 3 ] , i.e., the number of functions f : [ 8 ] [ 3 ] such that f ( 1 ) f ( 2 ) ≤ ··· ≤ f ( 8 ) . (c) the number of strictly increasing functions from [ 4 ] to [ 7 ] , i.e., the number of functions f : [ 4 ] [ 7 ] such that f ( 1 ) < f ( 2 ) < ··· < f
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