Combinatorics term in big O complexity

Question

Is it acceptable or best practice to have something like ${x \choose y}$ as a term in a big O complexity of an algorithm, or should that be simplified out further (to something like $x^y$)?

Answer 1

You should simplify it out. For example, if I were to tell you that for some $f(n)$ for large $n$, $$ f(n) = O\left( \binom{3n}{n} \right) $$ that would be a meaningful statement, but not very useful in terms of comparing how fast this function grows to other functions.It would be better to say that $$ f(n) = O\left( \left(\frac{3}{e}\right)^{2n}3^nn^{2n} \right) $$