three binomial(n,k) definitions

[Brendan McKay]

One can have fun discussing how to define binomial(n,k)
when k is not a non-negative integer.  However, modern
mathematics will crumble, libraries will self-destruct
and mothers will stop loving their children if a 
definition like 

   binomial(n,k) = if(k<0|k>n,0,n!/(k!*(n-k)!))

is adopted.  The reason is that the BINOMIAL THEOREM
requires
   binomial(n,k) = n*(n-1)*...*(n-k+1) / k!
for all non-negative integers k, REGARDLESS of n and even
if n is a complex number!  Vast tracts of mathematics,
including all the analysis that ultimately derives from
Taylors theorem, books full of combinatorial enumeration,
etc, etc, etc, ETC, depend on this.

So let's not get too carried away here!

Brendan.