Fermat's Combinatorial Theorem: prove, using comb. argument

[JackDaniels]

Hi there, I have been banging my head against this problem for days...

The following identity is known as Fermat's Combinatorial Identity.

( n ) = SUM from i=k to n of (i-1)
( k ) (k-1)

give a combinatorial argument to establish this identity.

I know that ( n ) = (n-1) + (n-1)
( r ) (r-1) ( r ) , and that a way of thinking about this equation is that there are

(n-1) (n-1)
(r-1) groups with one chose object in it, and ( r ) groups without that same object.... but I am having trouble relating that knowledge to the above question.

Thanks in advance.

JD

 

[JackDaniels]

Re: Fermat's Combinatorial Theorem

Oops, those equations didn't come out correctly....

n items take k would be

(n)
(k)

and the sum has i-1 items take k-1 inside it....