The simple counts. I can make Mathematica do them (using CS notation, P(n,r) and C(n,r)). Is there a reason that these don't exist as simple built-in functions (or can I just not find them)?
The function C is called Binomial in Mathematica. The function P does not appear to be pre-installed, but you can define it yourself:
P[n_, r_] := n!/(n - r)!
Binomial makes sense.
That's what I figured for Perm.
You might also take a look at Pochhammer and FactorialPower.