Splitting a team and scoring the products

A trading desk of 25 people stands together in one big group. You repeatedly pick any group that has at least two people and split it into two smaller nonempty groups. Every time you split a group of size $a$ into pieces of sizes $x$ and $a - x$, you write down the product $x \cdot (a - x)$ as your score for that split. You keep splitting until all 25 people stand alone, in groups of one.

Different people will split in different orders and different places, producing very different lists of scores along the way.

Prove that the grand total of all the scores you wrote down is always the same, no matter how the splitting is carried out, and find that total.