CF626D.Jerry's Protest

普及/提高-

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题目描述

Andrew and Jerry are playing a game with Harry as the scorekeeper. The game consists of three rounds. In each round, Andrew and Jerry draw randomly without replacement from a jar containing nn balls, each labeled with a distinct positive integer. Without looking, they hand their balls to Harry, who awards the point to the player with the larger number and returns the balls to the jar. The winner of the game is the one who wins at least two of the three rounds.

Andrew wins rounds 1 and 2 while Jerry wins round 3, so Andrew wins the game. However, Jerry is unhappy with this system, claiming that he will often lose the match despite having the higher overall total. What is the probability that the sum of the three balls Jerry drew is strictly higher than the sum of the three balls Andrew drew?

输入格式

The first line of input contains a single integer nn ( 2<=n<=20002<=n<=2000 ) — the number of balls in the jar.

The second line contains nn integers aia_{i} ( 1<=ai<=50001<=a_{i}<=5000 ) — the number written on the ii th ball. It is guaranteed that no two balls have the same number.

输出格式

Print a single real value — the probability that Jerry has a higher total, given that Andrew wins the first two rounds and Jerry wins the third. Your answer will be considered correct if its absolute or relative error does not exceed 10610^{-6} .

Namely: let's assume that your answer is aa , and the answer of the jury is bb . The checker program will consider your answer correct, if .

输入输出样例

  • 输入#1

    2
    1 2
    

    输出#1

    0.0000000000
    
  • 输入#2

    3
    1 2 10
    

    输出#2

    0.0740740741
    

说明/提示

In the first case, there are only two balls. In the first two rounds, Andrew must have drawn the 22 and Jerry must have drawn the 11 , and vice versa in the final round. Thus, Andrew's sum is 55 and Jerry's sum is 44 , so Jerry never has a higher total.

In the second case, each game could've had three outcomes — 10210-2 , 10110-1 , or 212-1 . Jerry has a higher total if and only if Andrew won 212-1 in both of the first two rounds, and Jerry drew the 1010 in the last round. This has probability .

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