There are $2010$ boxes labeled $B_1, B_2, . . . . B_{2010}$ and $2010n $ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves; each of which consists of choosing an $i$ and moving exactly $i$ balls from box $B_i$ into any one other box.
For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?
I'm unable to completely understand the question. There are (2010n)C(2010) ways of choosing the balls to keep them in each of the boxes. If the balls are identical, the permutation is not held into account. So, if we initially put n balls in each of the 2010 boxes, then the total number of balls used is 2010n. So, isn't it that for every value of n, there's 1 possibility that each box contains n number of balls? However, I'm unable to find out how many moves are needed when there is unequal distribution of balls as it happens in rest of the (2010n)C2010 - 1 cases.