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Math 184A
Final Exam
8–11AM Monday 12/13/99
1. AUTOMATIC contains 2 pairs of repeated letters and 7 distinct letters for a total of
9 letters. For 4
≤
n
≤
7, ﬁnd a formula for the number of
n
letter “words” that can
be made from this collection of letters. Your answer should be a formula involving
n
,
not 4 separate numbers for
n
=4
,
5
,
6
,
7.
2. Let
P
k
(
n
) be the number of permutations of the set
{
1
,...,n
}
having no cycles of
length greater than
k
.Thu
s
P
1
(
n
) = 1 for
n>
0 since all cycles are of length 1. For
later convenience, deﬁne
P
k
(
n
)=
‰
0
,
if
n<
0
1
,
if
n
=0.
(a) By considering the cycle containing
n
+1, prove that
P
2
(
n
+1) =
P
2
(
n
)+
nP
2
(
n

1)
for
n
≥
0. (Be careful for small values of
n
.)
(b) State and prove a similar recursion for
P
3
(
n
+ 1).
3. Deﬁne a “special” tree to be a rooted plane tree which is either
•
a single vertex (the root) or
•
a root vertex that is joined to either a left tree or a right tree or both a left and
right tree.
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This note was uploaded on 06/18/2009 for the course MATH 184a taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff
 Math

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