In class today a student had a question in regards to the negative binomial distribution, in particular about
(y-1) choose (r-1). She asked how to calculate this by hand. This is a combination and follows the same formula, n!/r!*(n-r)! This is because the distribution does not care where the success and failures fall in the sequence of y-1 and thus calculates the number of different COMBINATIONS for the string. To answer the question it will be (y-1)!/(r-1)!*(y-r)! *NOTE&* the y-1-(r-1)! = y-r!
Thanks for the nice comments for the question, Logan!
This is certainly a right explanation in terms of the calculation as well as the idea behind it.
I was thinking about an intuitive reason for the combination part but did not reach a proper way to deliver it.