Ok, it's been communicated that using the entropy function would provide a better approximation to nCr. So here is such a formula:

Let H(eps) := -eps.log_2(eps)-(1-eps)log_2(1-eps) be a binary entropy function.

Then binom(n, eps.n) is approximately equal to 2^(n.H(eps)).

Where eps is short for epsilon.

Update: Note that this uses the more common Stirling formula which is slightly less accurate as it truncates terms in the series (instead of approximating them like the formula i posted before).