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MAA AMC
AMC 12/AHSME
2021 AMC 12/AHSME Fall
23
23
Part of
2021 AMC 12/AHSME Fall
Problems
(1)
Quintuplet of integers
Source: 12B #23
11/17/2021
What is the average number of pairs of consecutive integers in a randomly selected subset of
5
5
5
distinct integers chosen from the set
{
1
,
2
,
3
,
…
,
30
}
\{ 1, 2, 3, …, 30\}
{
1
,
2
,
3
,
…
,
30
}
? (For example the set
{
1
,
17
,
18
,
19
,
30
}
\{1, 17, 18, 19, 30\}
{
1
,
17
,
18
,
19
,
30
}
has
2
2
2
pairs of consecutive integers.)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
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′
>
(
A
)
<
/
s
p
a
n
>
2
3
<
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c
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a
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=
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x
−
b
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>
(
B
)
<
/
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p
a
n
>
29
36
<
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c
l
a
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x
−
b
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>
(
C
)
<
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>
5
6
<
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p
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c
l
a
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=
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e
x
−
b
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>
(
D
)
<
/
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p
a
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>
29
30
<
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p
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c
l
a
s
s
=
′
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x
−
b
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>
(
E
)
<
/
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p
a
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>
1
<span class='latex-bold'>(A)</span>\ \frac{2}{3} \qquad<span class='latex-bold'>(B)</span>\ \frac{29}{36} \qquad<span class='latex-bold'>(C)</span>\ \frac{5}{6} \qquad<span class='latex-bold'>(D)</span>\ \frac{29}{30} \qquad<span class='latex-bold'>(E)</span>\ 1
<
s
p
an
c
l
a
ss
=
′
l
a
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x
−
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>
(
A
)
<
/
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p
an
>
3
2
<
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x
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>
(
B
)
<
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p
an
>
36
29
<
s
p
an
c
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a
ss
=
′
l
a
t
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x
−
b
o
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d
′
>
(
C
)
<
/
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p
an
>
6
5
<
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p
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c
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a
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=
′
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a
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x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
30
29
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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p
an
>
1
AMC
AMC 12
AMC 12 B