They explicitly write $nCr$ as $[n]C[r, n-r]$. This makes it better for "future uses", where it explicitly allows us to think of $[n]C[x, y]$ as breaking $n$ into $x$ things we choose and $y$ things we don't choose.

This makes the recurrence:

look as:

That is, we are reducing on either the first component ( $r-1$) or on the second component ( $n-r-1$), in the smaller set ( $n-1$).