*Post by HenHanna**Post by Jeff Barnett*Python says: (1 Combination 2) = 0

Ok... It's Impossible (to do).

------- is there a Better explanation?

(5 Combination 0) = 1 <---- This is explained by Comb(5,0)=Comb(5,5)

Comb(N,r)=Comb(N,N-r)

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from math import comb

for i in range(6): print( comb(5,i) )

print( comb(1,2) )

Let combination of n things taken m at a time be represented by [n,m].

Then [n,m] = [n,n-m] as you correctly note above. Further, we have the

computational formula [n,m] = n!/(m!(n-m)!) where x! is simply x

factorial. So [1,2] = 1!/(2!((-1)!)), or 1/2 divided by (-1)!. However

factorial of a negative integer is, by convention, an infinite value

so [1.2] = 0.

THank you...

Bard.Google.com says that

Comb(1,2) is not defined

factorial(-1) is not defined

factorial(-2) is not defined

GammaFunction(-1) is not defined

GammaFunction(-2) is not defined

They are partially correct. However, one can say that 1/0 = oo is

defined (where oo is infinity). In particular. 1/oo = 0 certainly makes

sense and that's all we need to accept for the above deductions.

Something I forgot to say in my original above is why x! = oo when x is

a negative integer. I cure that omission now. We want to define the

factorial as 1! = 1 and n! = n*(n-1)! when n > 1. We would also like to

be able to pedal backwards, i.e.. derive (n-1)! from n! and n. This is

certainly straightforward when n > 0. However the cases for other

integer n is trickier. For example, from our recursion formula we have

0! = 0*(-1)! and we know that (by definition) 0! =1. Thus, 1 = 0*(-1)!

which only has the possible symbolic solutions (-1)! = oo or -oo. Now a

trivial induction argument will draw the same conclusion for all

negative integers.

The importance of this bit of sophistry is our desire to do symbolic

manipulations with various classes of formulas without having to

numerically separate out a lot of special cases. Look at the above

definition of combination. With the established conventions, we have 1)

[n,n] = 1 and this can't work unless 0! = 1; 2) [n.0] = 1 which says

that the only o element subset of n elements is the empty set; etc.

It is a goal of mathematicians to make their definitions work not only

for the obvious cases but where there is darkness in our knowledge that

might be trivially illuminated by formulas that allow straightforward

consistent treatment throughout.

--

Jeff Barn