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Assume $\alpha + \beta = 1$, and one is trying to find a lower bound on the $k$th term in the binomial expansion of $(\alpha + \beta)^m$. The terms are of the form $\binom{m}{k} \alpha^k \beta ^{m-k}$.

Assume that $k \ll m \ll 1/\alpha$. More formally, and in big-O notation, $k = o(m)$ and $m = o(1/\alpha)$. Can we derive good lower bounds for the $k$th term in the binomial expansion? (When $m$ is large enough.)

PS 1: I read upper and lower bounds to binomial coefficient on PlanetMath, but couldn't figure out what to do.

PS 2: In fact, this question stem from the following (rather tight) inequality for large enough $n$'s:

$\binom{n^4}{n} \left(\frac{1}{n^3}\right)^n \left(1-\frac{1}{n^3}\right)^{n^4-n} < \frac{1}{4 \sqrt n}$

in which $m = n^4$, $\alpha=1/n^3$, and $k=n$.

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@Qiaochu: Thanks. I thing Stirling does the trick! –  Sadeq Dousti Jan 31 '11 at 2:36

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