How can I determine asymptotic growth of binomial coefficients?

Say I have a binomial coefficient $y=\binom{5n+3}{n+2}$ or $y=\binom{n^2+4}{3n}$ something of the sorts in terms of the variable $n$. How can I determine $f$ so that $y = O(f)$?

Is there a general method for this sort of thing. I have only ever seen $O$-notation in a computer science setting and its treatment was not concerned with deriving anything except the bounds for "simple" functions like logarithms and polynomials. As a side question, what would be a good resource to learn further methods to find $O$ of more complex functions? I understand the formal definition, but Im unsure how to use it practically.

My apologies if this is a stupid question. Its not homework.

• It's imprecise to say "determine O(y)." What you want is to write down some nice function f such that y = O(f). (Of course what you really want is probably to write down some nice function f such that y = Theta(f).) – Qiaochu Yuan Nov 7 '10 at 20:35
• You might also want to take a look at this question: math.stackexchange.com/questions/6784 – Mike Spivey Nov 7 '10 at 20:47
• @qiaochu yes, thats what I meant, edited. – AnonymousCoward Nov 7 '10 at 23:11

Use Stirling's approximation in $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.