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Permuted average (Posted on 2020-09-01)

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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analytic solution

| Comment 2 of 4 |

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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