Combinatorics

So now SpeedCrunch has got combinatorics support. Ariya had already implemented the famous ⁿC_r function, so I decided to code its cousin ⁿP_r.
I'd realized a nice optimization in Ariya's code. The general definition of ⁿC_r is given by

ⁿC_r = n! / ((n-r)! r!)
n, r ∈ N₀ , r ≤ n (applies all bellow)

But we can indeed reduce very much the number of integer multiplications with the derivation

r ∈ {0, n} ⇒ ⁿC_r = 1
r = 1 ⇒ ⁿC_r = n
r ∈ ]0, n/2] \ {1} ⇒ ⁿC_r = (r+1)_n / (n-r)!
r ∈ ]n/2, n[ \ {1} ⇒ ⁿC_r = (n-r+1)_n / r!

Note that (r+1)_n, for instance, is a modern factorial notation equivalent to (n)(n-1)...(r+1). I didn't know about this myself until now. This fasts computation a lot, else the very same calculations would be uselessly repeated.
In the very same way, I found a nice simplification for ⁿP_r. The general definition of ⁿP_r is given by

ⁿP_r = n! / (n-r)!

And again we can do some magic

r = 0 ⇒ ⁿP_r = 1
r = 1 ⇒ ⁿP_r = n
r = n ⇒ ⁿP_r = n!
r ∈ ]1, n[ ⇒ ⁿP_r = (n-r+1)_n

I spent some minutes getting to this conclusion and then I read it somewhere right after. First I felt a bit frustrated, then I smiled. It's funny to develop this program, it's funny to play maths.

I then realized my old Casio FX-880P can compute up to 69! and SpeedCrunch currently up to 96!, that's good enough (150-digit long integer). Nevertheless, when the user tries to calculate factorials, combinations or permutations with huge parameters the program freezes (even by the calc-as-you-type feature). Maybe it's really a good idea to define an upper limit for those functions input.