Binomial and NegBin distribution

Hi Folks.

I was hoping someone could help me understand this please.

Basically I need to find a relationship between the PMFs for the above. I can see the obvious similarities between the two PMFs and can explain them but an algebraic relationship eludes me.

I found some info online that says:

fXn(x) = fNx(n) + fMn-x(n)

where Xn is the number of successes in n trials; Nx is the number of trials to get x successes and Mn-x is the number of trials to get (n-x?) failures.

For info, the NegBin I'm interested in is the PMf [(x-1)!/(n-1)!(x-n)!]*p^n*(1-p)^x-n

All help appreciated. Thanks.

Dave.