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 $a>b>c$.

Without further restrictions on $a,b,c$ neither inequality can be correct:
Let $l(a,b,c)$ the LHS and $r(a,b,c)$ the RHS, then
$$l(3,2,1)=1/3 < r(3,2,1) = \exp(-2/3) \approx 0.513417$$
but
$$l(3,2,-1/2)=35/24 \approx 1.458333 > r(3,2,-1/2) = \exp(1/3)  \approx 1.395612$$ 
