Tourist Problem
题意翻译
给定一条直线上n个点的坐标$a_1,a_2,...a_n$，一条路线从原点开始，经过$a$的一个排列并在最后一个点结束（也就是说不返回原点）。一条路径的长度定义为排列中相邻两点$x,y$的距离$|x-y|$（包括原点），求所有路线的平均长度，要求化为最简分数。
题目描述
Iahub is a big fan of tourists. He wants to become a tourist himself, so he planned a trip. There are $ n $ destinations on a straight road that Iahub wants to visit. Iahub starts the excursion from kilometer 0. The $ n $ destinations are described by a non-negative integers sequence $ a_{1} $ , $ a_{2} $ , ..., $ a_{n} $ . The number $ a_{k} $ represents that the $ k $ th destination is at distance $ a_{k} $ kilometers from the starting point. No two destinations are located in the same place.
Iahub wants to visit each destination only once. Note that, crossing through a destination is not considered visiting, unless Iahub explicitly wants to visit it at that point. Also, after Iahub visits his last destination, he doesn't come back to kilometer 0, as he stops his trip at the last destination.
The distance between destination located at kilometer $ x $ and next destination, located at kilometer $ y $ , is $ |x-y| $ kilometers. We call a "route" an order of visiting the destinations. Iahub can visit destinations in any order he wants, as long as he visits all $ n $ destinations and he doesn't visit a destination more than once.
Iahub starts writing out on a paper all possible routes and for each of them, he notes the total distance he would walk. He's interested in the average number of kilometers he would walk by choosing a route. As he got bored of writing out all the routes, he asks you to help him.
输入输出格式
输入格式
The first line contains integer $ n $ ( $ 2<=n<=10^{5} $ ). Next line contains $ n $ distinct integers $ a_{1} $ , $ a_{2} $ , ..., $ a_{n} $ ( $ 1<=a_{i}<=10^{7} $ ).
输出格式
Output two integers — the numerator and denominator of a fraction which is equal to the wanted average number. The fraction must be irreducible.
输入输出样例
输入样例 #1
3
2 3 5
输出样例 #1
22 3说明
Consider 6 possible routes:
- \[2, 3, 5\]: total distance traveled: |2 – 0| + |3 – 2| + |5 – 3| = 5;
- \[2, 5, 3\]: |2 – 0| + |5 – 2| + |3 – 5| = 7;
- \[3, 2, 5\]: |3 – 0| + |2 – 3| + |5 – 2| = 7;
- \[3, 5, 2\]: |3 – 0| + |5 – 3| + |2 – 5| = 8;
- \[5, 2, 3\]: |5 – 0| + |2 – 5| + |3 – 2| = 9;
- \[5, 3, 2\]: |5 – 0| + |3 – 5| + |2 – 3| = 8.
The average travel distance is  =  = .