I'd like to prove that, for any $M,N\in\mathbb{N}$ with $M\leq N$, and any $n\in\mathbb{N}$ with $n\leq M$, the sum: $$\sum\limits_{k=0}^n\frac{\binom{M}{k}\!\!\binom{N-M}{n-k}}{\binom{N}{n}}=1.$$ I have proved it for $n=2$ by hand, and got stuck on the induction because trying to extract the previous sum is always unsuccesful. Let me expand on that. I tried twice, and always got stuck. I'll tell you only about the first time, because it is the attempt that tried to go from the induction hypothesis to the thesis, and it lays out all the equalities used in the second attempt, which went the other way, and was also unsuccessful. The induction step is: $$\sum\limits_{k=0}^{n-1}\frac{\binom{M}{k}\!\!\binom{N-M}{n-k-1}}{\binom{N}{n-1}}=1.$$ Using that the following holds: $$\binom{N}{n}=\frac{N!}{n!(N-n)!}=\frac{N-n+1}{n}\binom{N}{n-1}\!\!\implies\!\!\!\!\binom{N}{n-1}=\frac{n}{N-n+1}\binom{N}{n},$$ one turns the inductive hypothesis into: $$\sum\limits_{k=0}^{n-1}\frac{\binom{M}{k}\!\!\binom{N-M}{n-k-1}}{\binom{N}{n}}\frac{N-n+1}{n}=1.$$ This first step is lucky since the extra factor doesn't depend on $k$, so can be extracted from the sum and brought to the right side. The next step, though, is less lucky. One can easily see that: $$\binom{N-M}{n-k}=\frac{(N-M)!}{(n-k)!(N-M-n+k)!}=\frac{N-M-n+k+1}{n-k}\binom{N-M}{n-k-1}\!\!\implies$$ $$\implies\!\!\!\!\binom{N-M}{n-k-1}=\frac{n-k}{N-M-n+k+1}\binom{N-M}{n-k}.$$ This shows us how going on like this would give us a factor that can't be taken outside the sum, and can therefore not be isolated to get to the sum of the thesis (Note: "thesis" is what I have to prove, so the statement for $n$; I'm not sure if that term is used in English with that sense; it is in Italian). I tried using: $$\binom{N-M}{n-k}=\!\!\binom{N-M-1}{n-k-1}+\!\binom{N-M-1}{n-k}$$ as well, but that didn't get me any further. If we use the second equality above to substitute, we get: $$\sum\limits_{k=0}^{n-1}\frac{\binom{M}{k}\binom{N-M}{n-k-1}}{\binom{N}{n}}\frac{N-n+1}{n}\frac{n-k}{N-M-n+k+1}=1.$$
Any ideas?
