The University of Sydney
School of Mathematics and Statistics
Solutions to Practice Class 2
MATH2069/2969: Discrete Mathematics and Graph Theory
Semester 1, 2010
For some questions with a numerical answer, the answer is indicated in two forms, e.g.
12
2
= 6
. This is to give an indication of one way to obtain the answer; in a quiz, either
form on its own would suﬃce, e.g.
12
2
alone or
6
alone.
1.
Complete the Inclusion/Exclusion formulas:
(a)

A
∪
B

=

A

+

B
 − 
A
∩
B

(b)

A
∪
B
∪
C

=

A

+

B

+

C
 − 
A
∩
B
 − 
A
∩
C
 − 
B
∩
C

+

A
∩
B
∩
C

2.
Complete the table giving the number of ways to make an unordered selection of
k
things from
n
possibilities:
repetition allowed
(
n
+
k

1
k
)
repetition not allowed
(
n
k
)
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3.
Count the subsets of the set
{
A,B,C,D,E,F,G
}
:
(a)
which have 3 elements
(
7
3
)
= 35
(b)
which have 3 elements and contain C
(
6
2
)
= 15
(c)
whose intersection with
{
A,B,C,D
}
has size 2
(
4
2
)
×
2
3
= 48
4.
How many ways are there to select, from a barrel of 45 numbered balls:
(a)
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 Spring '09
 Math, Statistics, Graph Theory, Harshad number, A, University of Sydney School of Mathematics

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