24. $2 n$ points are uniformly distributed on a circle, and $n$ non-intersecting chords are used to pair these $2 n$ points into $n$ pairs. Find the number of all possible pairing methods.
24. Let the number of ways to pair $2n$ points be $f_{n}$. By selecting a point $P$, $P$ must be connected to a point $Q$ such that there are an even number of points on both sides of the line $PQ$. This leads to $f_{n}=f_{0} f_{n-1}+f_{1} f_{n-2}+\cdots+f_{n-1} f_{0}, f_{0}=1$.
Let $g_{n}=f_{n-1}$, then $g_{n+1}=g_{1} g_{n}+g_{2} g_{n-1}+\cdots+g_{n} g_{1}\left(g_{1}=1\right)$.
Thus, $g_{n}=\frac{1}{n} \mathrm{C}_{2 n-2}^{n-1}$ (see Example 2.4.6), and $f_{n}=\frac{1}{n+1} \mathrm{C}_{2 n}^{n}$.