Sum of Scores for Permutations
Consider a random permutation of the values 1, 2, 3, 4, 5, and 6, for example (1, 2, 3, 4, 6, 5). Let a distance of a number be defined as the number of places between the number's position from the position indexed by the number. For example, the distance of 6 in (1, 2, 3, 4, 6, 5) is 1 because 6 is in the 5th position, which is 1 place away from the 6th position. The score of a permutation is the total distance of all numbers in the permutation. In the example (1, 2, 3, 4, 6, 5), the score would be 2 = 1 + 1 for 5 and 6. Find the total score across all permutations of the values 1, 2, 3, 4, 5, and 6.
The answer is 8400.
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A number x is at position y in 6 6 ! permutations and it's distance is ∣ x − y ∣ . Since the contribution of a number to the score in a particular case is always independent of the position of other numbers, so we can add the distance of each number individually. Therefore the score is 6 6 ! ∑ x = 1 6 ∑ y = 1 6 ∣ x − y ∣ = 8 4 0 0