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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)
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(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)
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(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)
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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)
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(iii ) Apply the Karnaugh map method to find
a simpler Boolean expression for f .
(2 marks)
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(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)
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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 .
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(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)
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8004A Semester 2 2010
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8004A Semester 2 2010
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End of Extended Answer Section
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