nCr and nPr: Order of Operations Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
Where would a counting concept like nCr and nPr fall into this mix?

$_nC_r$ and $_nP_r$ are more like counting functions that map $\mathbb N^2\to \mathbb N$. What this means is they aren't operations more or less. they have an equation related to them $P(n,r):= \frac{n!}{(n-r)!}$ and $C(n,r):= \frac{n!}{r!(n-r)!}$. 
So in a sense they use multiplication, since $n!= 1*2*3*\dots*(n-1)*n$ . However algebraically, there is no relationship to the order of operations. 
