[2] [2] [2] [2] [2] 1. Short answer questions. Provide brief justification for your answers. (a) In how many ways can 51 red marbles and 37 blue marbles be distributed to 13 children if each child is to receive at least two blue marbles and at least 3 red marbles? (b) Let A = {0, 1, 2, 3,..., 9. How many 7-digit passwords consisting of numbers from A have a least fours 1's? Justify your answer. (c) How many terms are there in the expansion of the expression (x + y + z+w) ? (d) Draw an example of a graph that contains a cycle and a vertex of degree 1. (e) In how many ways can 25 distinct species of seeds be be planted in a garden with 17 rows so that in each row at least one seed is planted? [4] 2. Solve the recurrence an - 3an_1- 10an 2 = 0 for n 2 with a 0, a = 3. 3. Recurrence Relations and Generating functions: [4] (a) Consider the unique sequence a0, a1, a2,... satisfying an=6an 19an_2 for n 2 and a02, a = 9, anxn, then find the ratio- [2] Represent the sequence a0, a1,... by the generating function A(x) = nal form of this generating function by using the recurrence relation. (b) Calculate [n] ? + 3 X x3 ? x)2 ?n=0 4. Probability questions. For each part provide justification for your calculations. Suppose we select a binary strings of length 14 at random. Let X (s) be the number of 1 bits in s. [2] (a) What is Pr(X = 7)? [2] (b) What is Pr(X=7|X > 2)? [4] (c) Use the definition of expected value to show that E(X) = 7.