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Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Where would a counting concept like nCr and nPr fall into this mix?

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    The nCr and nPr are not directly related to probability if not combinatorics, that is, they are counts. – 2017-02-27
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    I edited it... thanks for pointing that out. – 2017-02-27

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There is no strong convention for these notations.

In most cases the $C$ and $P$ notations are self-delimiting, because the arguments are typeset as subscripts or superscripts. They can be written as ${}_nC_r$ or ${}^nC_r$ or $C^n_r$, where in each case it is unambiguous that the argument expression is whatever is written in smaller type -- or as $C(n,r)$ where the parentheses and comma make the structure explicit.

There are a few cases where ambiguity does arise, such that if you write the functions as ${}_nC_r$ and want to multiply two of them. In that case it is strongly advisable to use parentheses to disambiguate rather than try to rely on a convention: $({}_aC_b)({}_dC_e)$.

In any case, in more complex situations it is often preferred to use binomial-coefficient notation $\binom nr$ instead of ${}_nC_r$, and that is completely self-delimiting. The counts that are written ${}_nP_r$ in some elementary combinatorics texts appear to be rare enough in advanced material that it is generally feasible to write $\frac{n!}{(n-r)!}$ for them.

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$_nC_r$ 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.