Set of solutions for a binomial inequality I bumped into the following inequality:
$${a-b\choose c}{a\choose c}^{-1} \le \exp\left(-\frac{bc}{a}\right)$$
Playing with it a little bit, trying to bound it asymptotically for large $a$'s, using Stirling's approximation, I ended up with nothing. Finally I decided to put some numbers and check it, and figured out it is wrong. Moreover: it looked like it is always wrong.
Can you prove this inequality?
Edit: since it wasn't clear, I'll add that $a,b,c$ are all positive integers.
Edit: I also forgot to add the assumption $a>b>c$.

Actually, I don't know if you want LHS $\ge $ RHS or LHS $\le $ RHS. 
But here is an example: If you take $a=b+1=c+2$ --- e.g. $a=5,b=4,c=3$, then LHS $=0$ and the RHS is positive. So LHS $<$ RHS.
I think that given your restrictions $a>b>c$, it can never hold that LHS $\ge $ RHS.
