Identity using binomial theorem

Hi Guys first of all I would to thank in advance all people that read/try to solve this problem.

The problem is: prove that for all positive integer the following equality hold.

The only things that i was able to noting is if k=n we are done, because in the left-hand side is zero follow from the theorem

and on the right-hand site we try to select an k-element subsets form a n set where n il less that k so 0.

But im not able to generalize with binomial theorem (or by double counting).

Thanks a lot to all