MATH 681
Exam #1
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.
(10 points)
Prove the combinatorial identity
∑
n
k
=1
k
2
(
n
k
)
=
n
2
n

1
+
n
(
n

1)2
n

2
.
You may use any method you like.
2.
(10 points)
Answer the following questions relating to paths through the following
grid (note the excluded portions):
(a)
(5 points)
How many walks are there from the lower left corner to the upper
right corner taking upwards and rightwards steps only?
(b)
(5 points)
How many of these walks pass through the point marked with a solid
dot?
3.
(10 points)
Answer the following questions about the number of solutions
a
n
to the
equation
x
1
+
x
2
+
x
3
+
x
4
=
n
subject to the conditions that all the
x
i
are nonnegative
integers, that
x
1
≤
3, 2
≤
x
2
≤
4,
x
3
≥
5, and
x
4
≥
5.
(a)
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 Fall '09
 WILDSTROM
 Binomial, following questions, Recurrence relation, Generating function, ordinary generating function, combinatorial identity

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