MAD 2104
Oct 27, 2011
Quiz 5 Key
Prof. S. Hudson
1) [30pts] How many ways can 8 men and 5 women be placed in a line, such that no two
women stand next to each other ? Hint: place the men ﬁrst.
2) [20pts] In class, we proved a theorem about monotone subsequences. Using the notation
of that proof, ﬁnd (
d
3
,i
3
) for the sequence 3, 6, 9, 4, 7, 10, 5, 8, 11, 12.
3) [20pts] How many ways can you put 5 eggs into 3 baskets, A, B and C ?
4) [30pt] Choose ONE proof.
a) Determine which amounts of postage can be formed from 3 cent and 5 cent stamps.
Prove your answer using Strong Induction.
b) Give a combinatorial proof [Hint: count committeeswithaleader two ways] that
for all positive integers
n
,
n
X
k
=1
kC
(
n,k
) =
n
2
n

1
Remarks and Answers:
The average of the top 20 grades was approx 50, and the highest
was 70, which suggests this was a relatively hard quiz. The lowest scores were on Problem
1, which (I thought) was a fairly standard counting problem. The unoﬃcial scale is;
A’s 65  100
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 Fall '08
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 Mathematical Induction, 1 k, Strong Induction, 20pts, 30pts, 30pt

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