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Math 239: Quiz
Solutions
1. (5 points) Assume
r
and
n
are nonnegative integers. Express the coefﬁcient of
x
n
in the
following series as a summation:
(1

3
x
)
r
(1
+
5
x
2
)

10
.
Solution:
Using the binomial theorem we have
(1

3
x
)
r
=
X
i
≥
0
(

3)
i
ˆ
r
i
!
x
i
and
(1
+
5
x
2
)

10
=
X
j
≥
0
5
j
ˆ

10
j
!
x
2
j
.
Hence setting
i
=
n

2
j
and using the product rule, we get
[
x
n
](1

3
x
)
r
(1
+
5
x
2
)

10
=
X
0
≤
j
≤
n
/2
(

3)
n

2
j
5
j
ˆ
r
n

2
j
!ˆ

10
j
!
=
(

3)
n
X
0
≤
j
≤
n
/2
±
5
9
¶
j
ˆ
r
n

2
j
!ˆ

10
j
!
2. (5 points) At a bake sale, there is one cake for $12, two pies costing $6 and $8, and 10
mufﬁns at $1 each. Let
b
n
be the number of ways that you can spend $n. Find the gen
erating function
∑
n
≥
0
b
n
x
n
in the form of a rational function. Give full justiﬁcation for your
answer.
Solution:
Deﬁne the four sets
S
1
=
{0,12}, S
2
=
{0,6}, S
3
=
{0,8}, S
4
=
{0,1,.
..,10}.
Then our spending decisions correspond to the elements of
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 Spring '09
 M.PEI
 Math, Binomial Theorem, Integers, Binomial

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