On a circle, there are 20 points. These 20 points are connected by 10 chords, none of which share endpoints and do not intersect.

How many ways can this be done

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Let $a_{n}$ be the number of ways to connect $2 n$ points on a circle with $n$ non-intersecting chords. It is clear that $a_{1}=1$ and $a_{2}=2$. We will show that

$a_{n}=a_{n-1}+a_{n-2} a_{1}+a_{n-3} a_{2}+\ldots+a_{1} a_{n-2}+a_{n-1}$

Fix one of the $2 n$ points. The chord emanating from it divides the circle into two arcs, each containing an even number of the given points. If one arc contains $2 k$ points, then the other contains $2(n-k-1)$ points; these points can be connected by non-intersecting chords (not intersecting the first chord) in $a_{n-k-1} a_{k}$ ways. It remains to sum over $k$ from 0 to $2 n-2$.

Thus, $a_{3}=2 a_{2}+a_{1}^{2}=5, a_{4}=2 a_{3}+2 a_{1} a_{2}=14, a_{5}=2 a_{4}+2 a_{1} a_{3}+a_{2}^{2}=42, \ldots, a_{10}=$ 16796.

## Answer

16796 ways.

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