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8004A Semester 2 2010 Page 10 of 20 Extended Answer Section There are three questions in this section, each with a number of parts. Write your answers in the space provided below each part. If you need more space there are extra pages at the end of the examination paper. 1. A shop sells pencils of five colours: red, orange, yellow, green and blue. Mrs Schwarz goes to the shop, buys a selection of pencils and distributes them all to her children Ann, Ben, Chris and Don. What is the number of different outcomes under the condition that (a) Each of the children gets two pencils? (2 marks) (b) Each of the children gets three pencils of different colours? (2 marks) 8004A Semester 2 2010 Page 11 of 20 (c) Suppose Mrs Schwarz buys five pencils, one pencil of each colour. What is the number of different ways to distribute them to the children so that Ann and Ben get two pencils each, Chris gets one pencil and Don gets none? (2 marks) (d) Suppose Mrs Schwarz buys ten green pencils. What is the number of different ways to distribute them to the children? (2 marks) 8004A Semester 2 2010 Page 12 of 20 (e) Suppose Mrs Schwarz buys ten green pencils. What is the number of different ways to distribute them to the children so that at least two of the children get the same number of pencils? (2 marks) 8004A Semester 2 2010 Page 13 of 20 2. (a) Given propositions x, y and z, consider the compound proposition (x∨∼y) ⇒ (y∧z). (i ) Construct the truth table for this proposition. (2 marks) (ii ) Interpreting the proposition as a Boolean function f in the variables x, y and z (replace T by 1 and F by 0 in the table), write down the Boolean expression for f in disjunctive normal form. (2 marks) 8004A Semester 2 2010 Page 14 of 20 (iii ) Apply the Karnaugh map method to find a simpler Boolean expression for f . (2 marks) 8004A Semester 2 2010 Page 15 of 20 (b) For any real parameter a ∈ R consider the proposition p(a) given by � � � � ∀x ∃y xy = 0 ⇒ x 6= a , where the universal set U is the set of all real numbers. (i ) Apply the negation rule to the quantifiers to write ∼p(a) without using the implication sign ⇒. (2 marks) (ii ) Using your answer to the previous part or otherwise, find all values of a for which p(a) is true. (2 marks) 8004A Semester 2 2010 Page 16 of 20 3. (a) The sequence an is defined by the recurrence relation an = −8an−1 + 33an−2 , n � 2, subject to the initial conditions a0 = 1 and a1 = 17. (i ) Find an explicit formula for an . (3 marks) (ii ) Using the explicit formula for an found in the previous part or otherwise, find (2 marks) a closed form for the generating function of the sequence an . 8004A Semester 2 2010 Page 17 of 20 (b) Let xn and yn be two sequences. Suppose that they satisfy the respective recurrence relations xn = xn−1 −xn−2 + n2 and yn = yn−1 −yn−2 + n2 −2n+ 1 with n � 2. Write down the recurrence relation satisfied by the sequence zn defined by the formula (2 marks) zn = xn − yn . (c) The generating function B(z) of the sequence bn is given by the expression 1 + 2z . B(z) = 1 + z + z2 Give a closed form for the generating function C(z) of the sequence cn defined by the formula n � 0. cn = bn+2 , (3 marks) You may use next pages for your answers 8004A Semester 2 2010 Page 18 of 20 This page may be used if you need more space for your answers 8004A Semester 2 2010 Page 19 of 20 This page may be used if you need more space for your answers 8004A Semester 2 2010 Page 20 of 20 This page may be used if you need more space for your answers End of Extended Answer Section This is the last page of the question paper.