Score from splitting 1000 coins down to singletons

Begin with one heap of $1000$ coins. At each step choose a heap of size at least $2$, divide it into two heaps of sizes $a$ and $b$, and score $ab$ points. Continue until every heap holds one coin. What total score results?

$499{,}500$, whatever splits you make. Each split of $a$ and $b$ separates exactly $ab$ pairs of coins, and every pair of coins is separated exactly once, so the total is $\binom{1000}{2}=499500$. (Induction on $n$ shows the score is always $n(n-1)/2$.)