Coin Split Problem

The Problem

Prove by induction: if you split a pile of $n$ coins into two piles of $a$ and $b$ coins ($a+b=n$), and repeat this on each pile until all piles have 1 coin, then the total number of splits performed is exactly $n-1$, regardless of the splitting order.

Also: prove that ${\sum }_{\text{splits}}{a}_i\cdot b_i=n(n-1)/2$ (sum of products of pile sizes at each split).