Total Score of a Random Partitioning Game

You start with a single set containing $n$ items. At each step, you pick one of the current sets (of size $k \geq 2$) and randomly split it into two non-empty subsets of sizes $k_1$ and $k_2$ (where $k_1 + k_2 = k$). You score $k_1 \times k_2$ points for that split. You keep going until every set is a singleton. What is the total score at the end of the game? Show that the answer is the same no matter how the random partitions happen to fall.