Question

(6) (a) Consider the follow ing graph U T S 1] (ii) Does the graph have a closed Euler trail? If so, give an example of a closed Euler trail in 2] 1] (iv) Two identical looking bags are on a table. One cont ains 30 green marbles and 30 black marbles, and the other contains 10 green marbles, 10 blue marbles and 10 red marbles One of the bags is randomly selected (each has a 50% chance of being chosen) and a marble is drawn uniformly at random from that bag. Given that the marble drawn is green, what is the probability it comes from the bag with green and black marbles? (i) What are the degrees of the vertices in the graph? the graph. If not, explain why no closed Euler trail exists (iii) Give an example of a spanning tree in the graph. 6 (b) For each integer n 2 1, let rn be the number of binary strings of length n that do not contain three consecutive 1s. (i) Find r, , s and r (ii) Find a recurrence for rn that holds for all integersn 2 4. Explain why your recurrence gives n For each integer n 2 5, let sn be the number of binary strings of lengthn that do not contain thee consecutive ls, do not begin with 1 and end with two consecutive 1s, and do not begin with two consecutive 1s and end with 1 (iii Find an expression for sn in terms of ri, r2, ...,rn-1 that holds for all integers n 2 5. Explain why your expression gives sn. 2] [4] [4]

Answer 1

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