Sample Exam 3
EECS 203
Winter 2007
You have two hours to complete this exam. You may use any information you have
written on three 8
.
5”
×
11” sheets of paper, but no other information. Leave at least
one seat between yourself and other students.
Problem 1.
Zero or more of the following relations on the set of all UM undergraduates are
reﬂexive, symmetric,
and
transitive. Which ones are they?
(a)
R
=
{
(
x, y
)

x
and
y
have taken, or are taking, EECS 203
}
.
(b)
R
=
{
(
x, y
)

x
and
y
have the same last digit of their student ID number
}
.
(c)
R
=
{
(
x, y
)

x
’s GPA is greater than
y
’s GPA
}
.
(d)
R
=
{
(
x, y
)

x
and
y
are enrolled in the same college or live in the same dorm
}
.
Problem 2.
How many diFerent strings can be made from the word PEPPERER such that the
strings begin and end with diFerent letters? Zero or more of the following statements
are may be correct.
(a) 6!
/
(2!
·
2!
·
2!) + 2
·
6!
/
(2!
·
3!)
(b) 8!
/
(3!
·
3!
·
2!)
(c) 2(6!
/
(2!
·
2!
·
2!) + 6!
/
(2!
·
3!) + 6!
/
(3!
·
2!))
(d)
(
8
2
)(
6
3
)(
3
2
)
Problem 3.
Zero or more of the following statements are true. Which ones are they?
(a)
∑
n
k
=0
(
n
k
)(
3
2
)
k
=
(
5
2
)
n
(b)
∑
n
k
=0
k
(
k
−
1)
(
n
k
)
=
n
(
n
−
1)2
n
−
2
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∑
r
k
=0
(
m
r
−
k
)(
n
k
)
=
(
m
r
+
n
)
(d)
∑
n
k
=0
(
n
k
)
3
=
(
3
n
n
)
Problem 4.
Consider the directed graph below, where
L
indicates the
level
of the graph and
n
indexes the vertices at a given level. DeFne an
interior vertex
at a given level to
be any vertex with three inbound edges. ±inally, let
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 Winter '07
 YaoyunShi
 Equivalence relation, Transitive relation, 0 l, r1, Dogbert, n. Substitute

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