MATH 681
Final Exam
Answer exactly four of the following six questions.
Indicate which four you would like
graded!
Binomial coeﬃcients, Stirling numbers, and arithmetic expressions need not be simpliﬁed
in your answers.
1.
(15 points)
Answer the following questions about ﬁnding the number of words
a
n
of
length
n
with letters A, B, and C using the letter “A”
at least once
.
(a)
(5 points)
Find an exponential generating function for
a
n
.
(b)
(5 points)
Develop an argument to show that
a
n
satisﬁes the recurrence
a
n
=
2
a
n

1
+ 3
n

1
with
a
0
= 0.
(c)
(5 points)
Using a method of your choice, ﬁnd a closedform expression for
a
n
.
2.
(15 points)
Prove the following identities using combinatorial arguments, where
F
n
represents the Fibonacci sequence indexed with
F
0
= 1,
F
1
= 1,
F
2
= 2.
(a)
(5 points)
k
(
n
k
)
=
n
(
n

1
k

1
)
.
(b)
(5 points)
F
n
=
(
n
0
)
+
(
n

1
1
)
+
(
n

2
2
)
+
···
+
(
d
n
2
e
b
n
2
c
)
.
(c)
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 Fall '09
 WILDSTROM
 Binomial, Order theory, Recurrence relation, Generating function, ordinary generating function, Dilworth

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