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I'm looking for asymptotic estimate for the binomial coefficient: $ \ln{\binom{n}{[\sqrt{n}]}} $ I assume Stirling's approximation can help, but I'm not sure I will get any good estimation with this approach. Is there any good way to make an estimation for this coefficient? Thanks in advance.

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    Use $\ln(n!)=(n+1/2)\ln(n)-n+O(1)$ for all the three terms.2012-10-05

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Using Shitikanth's hint I think you're going to be coming up with $\text{ln}{n \choose [\sqrt{n}]}\approx\text{ln}\left(\frac{n^{n+\sqrt{n}/2+3/4}}{\left(n-\sqrt{n}\right)^{1/2-\sqrt{n}+n}}\right).$

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    It may just be that ${n \choose [\sqrt{n}]}$ is an important term in a lot of algorithms and that's why they put it in there. For what it's worth my answer looks right based on the plot of it: http://i.imgur.com/JOPFU.gif2012-10-06