Functions for permutations and combinations counting?

Questions
1

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)?

2

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)!
3

Binomial makes sense.
That’s what I figured for Perm.
Thanks.

4

You might also take a look at Pochhammer and FactorialPower.

5

That doesn’t answer the original question, which is one I have as well. Why aren’t these native to Mathematica?

6

Binomial (which counts combinations) and Pochhammer (which counts permutations) are both Mathematica system functions.

7

In[112]:= Pochhammer[3, 2]

Out[112]= 12

But 3P2 = 6, not 12

8

Check the reference page for Pochhammer to see how the arguments get used. In particular, to map to permutations (n,k), one would use Pochhammer[k,n-k+1].

9

Thanks, Daniel, I hadn’t thought to do that.

But for my own convenience, I will create a user-defined function as suggested by
Gianluca.

10

In my humble opinion, permutations (n,k) or nPk may be calculated by Pochhammer[n-k+1, k], not by Pochhammer[k,n-k+1]. For example, 5P2 is 5!/(5-2)! = 5!/3! = 5x4=20. Pochhammer[4,2] gives 20. Pochhammer[2,4] results in 120. I am just learning ropes in Mathematica and my math skills are quite rusty. Please, take my words with a grain of salt.
Sang