Math 184A
Final Exam (80 points)
8AM 7 Dec. 2005
•
Please put your name and ID number on your blue book.
•
CLOSED BOOK except for BOTH SIDES of one page of notes.
•
Calculators are NOT allowed.
•
You must show your work to receive credit.
1. (16 pts.) A square table has two seats on each side for a total of eight seats. Rotations
of the table don’t matter. Thus, if 1
,
2
, . . .,
8 are placed around the table,
1 2
8
3
7
4
6 5
and
7 8
6
1
5
2
4 3
are the same, but di±er from
2 1
3
8
4
7
5 6
and
8 1
7
2
6
3
5 4
.
(a) How many ways can eight people be seated at the table?
(b) We have four identical red chairs and four identical blue chairs. How many ways
can the eight chairs be placed around the table? Again, rotations of the table do
not matter.
2. (18 pts.) Let
V
=
{
1
,
2
, . . ., n
}
.
(a) Compute the number of simple graphs with vertex set
V
that have exactly
q
edges.
(b) A vertex is
isolated
if it does not lie on any edges. Suppose
S
⊂
V
. Compute the
number of simple graphs with vertex set
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 Fall '08
 staff
 Math, Graph Theory

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