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Permuted average (Posted on 2020-09-01) Difficulty: 3 of 5
For each permutation a1, a2, . . . , a10 of the integers 1,2,3,...,10, form the sum |a1 - a2| + |a3 - a4| + |a5 - a6| + |a7 - a8| + |a9 - a10|.

Find the average value of all such sums.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution analytic solution | Comment 2 of 3 |
There are 45 different ways to pick two numbers randomly from the set.  Their expected absolute difference is 
(1*9 + 2*8 + 3*7 + 4*6 + 5*5 + 6*4 + 7*3 + 8*2 + 9*1)/45 =
165/45 = 11/3

The average value from summing 5 absolute values across all permutations therefore = (5 * 11/3) = 55/3.

  Posted by Steve Herman on 2020-09-01 12:01:11
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