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Turkey Contests
National Olympiad First Round
2012 National Olympiad First Round
32
32
Part of
2012 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 2012 Problem - 32 {Combinatorics}
Source:
7/1/2012
How many permutations
(
a
1
,
a
2
,
…
,
a
10
)
(a_1,a_2,\dots,a_{10})
(
a
1
,
a
2
,
…
,
a
10
)
of
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
1,2,3,4,5,6,7,8,9,10
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
satisfy
∣
a
1
−
1
∣
+
∣
a
2
−
2
∣
+
⋯
+
∣
a
10
−
10
∣
=
4
|a_1-1|+|a_2-2|+\dots+|a_{10}-10|=4
∣
a
1
−
1∣
+
∣
a
2
−
2∣
+
⋯
+
∣
a
10
−
10∣
=
4
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
60
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
52
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
50
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
44
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
36
<span class='latex-bold'>(A)</span>\ 60 \qquad <span class='latex-bold'>(B)</span>\ 52 \qquad <span class='latex-bold'>(C)</span>\ 50 \qquad <span class='latex-bold'>(D)</span>\ 44 \qquad <span class='latex-bold'>(E)</span>\ 36
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
60
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
52
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
50
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
44
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
36