Question

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(6) (a) Consider the following graph P R U T (i) What are the degrees of the vertices in the graph? (ii) Does the graph have a closed Euler trail? If so, give an example of a closed Euler trail in the graph. If not, explain why no closed Euler trail exists. (iii Give an example of a spanning tree in the graph (iv) Two identical looking bags are on a table. One contains 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 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? a (b) For each integer n > 1, let r, be the number of binary strings of length n that do not contain three consecutive 1s. (i) Find rı, r2, r3 and r4 (ii) Find a recurrence for rn that holds for all integers n 2 4. Explain why your recurrence gives rn For each integer n 2 5, let s,n be the number of binary strings of length three consecutive 1s, do not begin with 1 and end with two consecutive 1s, and do not begin with two consecutive 1s and end with 1 n that do not contain (iii Find an expression for sn in terms of rı,r2, ..., rn-1 that holds for all integers n > 5. Explain why your expression gives sn

Answer 1

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