Math 552 Midterm 2 SOLUTIONS
Rob Bayer
July 27, 2009
1. (10 pts) Short answer. You need not show any work for this section
(a) How many ways are there to put
n
balls in
k
bins if:
i. The balls and the bins are both numbered
k
n
ii. The balls are indistinguishable and the bins are numbered
n
+
k

1
k

1
iii. The balls are numbered and the bins are indistinguishable
k
X
i
=1
S
(
n, i
)
iv. Both the balls and the bins are indistinguishable
p
k
(
n
)
(b) Find the coefficient of
x
8
in (5
x

7)
21
(
21
8
)
5
8
(

7)
13
(c) State the Generalized Pigeonhole Principle:
If you put
n
pigeons in
k
boxes, some box contains at least
d
n
k
e
pigeons
(d) Define what it means for a permutation to be a derangement:
The permutation has no fixed points. (ie, every element gets moved)
(e) Find the flaw in the following “proof” that if
a
6
= 0, then
a
n
= 1 for all nonnegative integers
n
:
•
BC (n=0)
:
a
0
= 1 since
a
6
= 0
•
IH
: Suppose the claim is true for all exponents
≤
k
. (Formally:
a
i
= 1 for all
i
≤
k
)
•
IS
:
a
k
+1
=
a
k
·
a
k
a
k

1
IH
= 1
·
1
1
= 1
There aren’t enough base cases–when trying to prove it for
a
1
(ie,
k
= 0) we need an
a

1
which we
don’t have a base case for. (In general, since we go down 2 steps in our IS, we need 2 base cases)
2. (15 pts) Prove that
1
1
·
3
+
1
3
·
5
+
1
5
·
7
+
· · ·
+
1
(2
n

1)(2
n
+ 1)
=
n
2
n
+ 1
for every
n
≥
1
We’ll go by induction on
n
:
•
BC (n=1)
The LHS is just
1
3
and the RHS is
1
2+1
=
1
3
•
IH
Suppose that
1
1
·
3
+
1
3
·
5
+
1
5
·
7
+
· · ·
+
1
(2
k

1)(2
k
+1)
=
k
2
k
+1
•
IS
1
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1
1
·
3
+
1
3
·
5
+
1
5
·
7
+
· · ·
+
1
(2
n

1)(2
n
+ 1)
+
1
(2
k
+ 1)(2
k
+ 3)
IH
=
k
2
k
+ 1
+
1
(2
k
+ 1)(2
k
+ 3)
=
k
(2
k
+ 3) + 1
(2
k
+ 1)(2
k
+ 3)
=
2
k
2
+ 3
k
+ 1
(2
k
+ 1)(2
k
+ 3)
=
(
k
+ 1)(2
k
+ 1)
(2
k
+ 1)(2
k
+ 3)
=
k
+ 1
2
k
+ 3
Thus, by induction, the result holds for all
n
≥
1
3. (15 pts) Consider the set
S
defined as follows:
•
001
∈
S
•
Whenever
w, v
∈
S
, 100
w
∈
S,
00
w
1
∈
S
and
wv
∈
S
Prove that every string in
S
ends with a 1 and contains twice as many 0’s as 1’s.
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 Summer '08
 STRAIN
 Math, Mathematical Induction, Want, pts, Negative and nonnegative numbers, Recurrence relation, Structural induction

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