Hint: Induction on k.
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
For and define this is a generalized binomial coefficient. Fix .
Define and .
We will show and it will follow that .
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Now,
Thus,
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Hmm, I am not exactly sure where the mistake is and it is too late for me to check.
They seem to be almost the same.
EDIT: There is no mistake we have shown
This means for and .
Thus, if and it works out.