Average number of swaps performed in Bubble Sort

Question

I came across this problem right now: http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=24&page=show_problem&problem=3155 The problems asks to calculate the average number of swaps performed by a Bubble Sort algorithm when the given data is a shuffled order of first n (1 to n listed randomly) natural numbers.

So, I thought that:

1. Max no of swaps possible=n(n-1)/2. (When they are in descending order.)
2. Min no of swaps possible=0. (When they are in ascending order.)

So, the mode of this distribution is (0+n(n-1)/2)/2 =n(n-1)/4. But, this turned out to be the answer. I don't understand why did the mode coincide with mean.

Answer 1

Since the inputs to be sorted can be any random number with equal probability of occurrence, the distribution is symmetrical.

It is a property of symmetrical distributions that their mean, median and modes coincide which is why the mean and mode are same.