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Math 184A
Final Exam (100 points)
21 March 2007
•
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. (18 pts.) Let
S
(
n,k
) be the Stirling numbers of the second kind — the number of
partitions of an
n
set into
k
blocks. Derive the following
(a)
S
(
n,
2) = 2
n

1

1
(b)
S
(
n,n

1) =
n
(
n

1)
2
.
2. (10 pts.) A long rectangular table has
n
seats on each side and none at the ends. We
want to seat
n
men and
n
women at the table so that no one is next to or opposite a
person of the same sex. How many ways can this be done?
Note
: There are no symmetries—the sides of the table are diﬀerent.
3. (14 pts.) Each edge of a simple graph
G
is assigned a weight from the set
{
1
,
2
,...,
9
}
and each of the nine weights is used at least once.
(a) If the graph
G
has 10 (ten) edges, what is the largest number of minimum weight
spanning trees it can have? Why?
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 Math

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