Catalan Definition Numbers

Question

I am currently taking discrete math 2, and we've just learned about Catalan Numbers.

In the formula given, the following was shown: ${2n \choose n}$

What does this mean, as far as these two numbers being in the same parenthesis together? I apologize, as I didn't really know what to ask, since I have no idea what this is called. Any help or pointing to resources would be much appreciated. Thank you!

Answer 1

This is called the binomial coefficient. In general, $$\binom nk = \frac{n!}{(n-k)!\,k!}.$$ It is the coefficient of $x^k$ in the expansion of $(1+x)^n$, as well as the number of ways of choosing $k$ out of $n$ objects, with repetition not allowed and order not important.

Answer 2

The binomial coefficient $ {\binom mk}=\frac{m!}{k!(m-k)!}$, so $\binom{2n}n=\frac{(2n)!}{(n!)^2}$