MAD 2104
June 3, 2010
Quiz 5 and Key
Prof. S. Hudson
1a) [20pts each] Suppose
k
integers are chosen from the ﬁrst eleven positive integers. Show
that if
k
≥
8, then some pair of the chosen integers has a sum equal to 12.
1b) Find the smallest value of
k
that makes the conclusion true. Justify your answer
brieﬂy.
2a) [15pts each] A coin is tossed 10 times, where each toss comes up heads or tails. How
many possible outcomes are there?
2b) How many are there, with at most 2 heads ?
3a) [15pts each] Let
R
=
{
(1
,
2)
,
(3
,
1)
,
(2
,
1)
,
(1
,
3)
,
(2
,
2)
}
be a relation on
A
=
{
1
,
2
,
3
}
.
Is
R
symmetric ?
3b) Is it transitive ? (justify brieﬂy)
Remarks and Answers:
The average among the top 20 students was about 65 / 100.
Here is a rough scale for the quiz, based mainly on that average:
As 75 to 100
Bs 65 to 74
Cs 55 to 64
Ds 45 to 54
1a) The pigeons are the
k
chosen integers. The pigeonholes are the 6 sets
{
1
,
11
}
,
{
2
,
10
}
,
{
3
,
9
}
,
{
4
,
8
}
,
{
5
,
7
}
and
{
6
}
. By the PHP (since
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 Adrenergic receptor, G protein coupled receptors, Adrenergic receptors, Prof. S. Hudson

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