Math 184A
First Exam
27 October 1999
1. Calculate the number of 6 card hands that contain
(a) two 3ofakind; e.g., 3 eights and 3 jacks;
(b) two pair, but not three pair or 3 of a kind.
2. Give an example of each of the following, or explain why no such example can exist.
(a) An ordered list of length 5 with
no repeats
chosen from a set of 4 elements.
(b) A 5 vertex, 5 edge simple graph that is not connected.
(c) A 4 vertex, 10 edge simple graph that is connected.
3. Recall that a
simple graph
with vertices
V
is (
V, E
) where the edges
E
are a
set
chosen
from
P
2
(
V
) and a simple directed graph (
) has edges chosen form
V
×
V
.
(a) A
simple multigraph
with vertices
V
is (
) where the edges
E
are a
multiset
chosen from
P
2
(
V
). Prove that the number of simple multigraphs with vertices
V
=
{
1
,...,n
}
and
q
edges is
(
N
+
q

1
q
)
where
N
=
(
n
2
)
.
(b) A
simple directed multigraph
with vertices
V
is (
) where the edges
E
are a
multiset
chosen from
V
×
V
. Find and prove a formula for the number of simple
directed multigraphs with vertices
V
=
{
1
,
2
}
and
q
edges.
4. A partition of a positive integer is an unordered list of positive integers whose sum is
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 Fall '08
 staff
 Math, Set Theory, Natural number, edge simple graph

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