Find the number of expressions containing $n$ pairs of parentheses which are correctly matched. For example, when $n=3$, the answer is $5$ because all the valid expressions are $$((())),\ (()()),\ (())(),\ ()(()),\ ()()()$$
The answer is $\boxed{\frac{1}{n+1}\binom{2n}{n}}$ by using the model in # 4355.
On the $n\times n$ grid, we can interpret the unit right step as the left parenthesis and the up step as the right parenthesis. Because it is a $n\times n$ grid, a path will have exactly $n$ pairs of parentheses. Meanwhile, these parentheses are correctly matched if and only if at any point there are no more right parentheses than the left ones, i.e. no more up steps than the right steps. This is equivalent to saying the path can only stay in the right $x\ge y$.