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2023 LMT Fall
21
2023 Fall LMT Speed p21 \sum^5_{i=1} |a_i -i |
2023 Fall LMT Speed p21 \sum^5_{i=1} |a_i -i |
Source:
March 2, 2024
algebra
number theory
combinatorics
Problem Statement
Let
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
(a_1,a_2,a_3,a_4,a_5)
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
)
be a random permutation of the integers from
1
1
1
to
5
5
5
inclusive. Find the expected value of
∑
i
=
1
5
∣
a
i
−
i
∣
=
∣
a
1
−
1
∣
+
∣
a
2
−
2
∣
+
∣
a
3
−
3
∣
+
∣
a
4
−
4
∣
+
∣
a
5
−
5
∣
.
\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.
i
=
1
∑
5
∣
a
i
−
i
∣
=
∣
a
1
−
1∣
+
∣
a
2
−
2∣
+
∣
a
3
−
3∣
+
∣
a
4
−
4∣
+
∣
a
5
−
5∣.
Proposed by Muztaba Syed
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