Is $n\binom{\epsilon n}{t}>t\binom{n}{t}$ for large $n$ and fixed $\epsilon$ and $t$ Let $\epsilon$ and $t$ be fixed numbers with $t$ and integer. I came across the following inequality in a counting problem. 
$$n\binom{\epsilon n}{t}>t\binom{n}{t}.$$
I want to show that for $n$ large enough this inequality is true. I don't know if my hypothesis is true or not. I've made many lists in mathematica with different $\epsilon$ and $t$. All of the lists showed my hypothesis was correct in that case. Any ideas on how to prove this result? Any much appreciated. 

Let $\epsilon>0$ and $n,t\in\mathbb{N}$. Assume for sake of contradiction that for all $n$
$$n\binom{\epsilon n}{t}\leq t\binom{n}{t}.$$
Then that would yield 
$$ \binom{\epsilon n}{t}/\binom{n}{t}\leq \dfrac{t}{n}.$$
for all $n$. Using the appropriate approximations from this wikipedia page 
 http://en.wikipedia.org/wiki/Binomial_coefficient
We have 
$$ \dfrac{\epsilon^{t}}{t^{t}t!}\leq \dfrac{t}{n}$$
for all $n$ which is clearly false because $\epsilon $ and $t$ are fixed.
