It was hard to understand how to calculate the answer.

It is relatively easy to invent a correct solution when you know the expected time complexity.

Solve the task online,  read elements one by one.
So,  you have dp[i][j] = count of different j-inversions ending at element number j.

dp[i] [j] =sum (dp[p] [j-1]) for all p>i.

The answer is sum(dp[i] [k])  for all i=0...n-1.

As mentioned below,  an array of Fenwick trees is your friend.