Hypergeometric Distribution - Combinatorial Identities

Combinatorial Identities

As one would expect intuitively, the probabilities sum up to 1 :

This is essentially Vandermonde's identity from combinatorics.

Also note that the following identity holds:

This follows clearly from the symmetry of the problem, but it can also be shown easily by expressing the binomial coefficients in terms of factorials, and rearranging the latter.