I know that the answer uses the Catalan numbers so C4*C7, but I'm not sure how you get to the 4 and 7? The question is referencing a circle with 24 points around it each labelled 1 - 24.
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The key idea is that the single chord divides the interior of the circle into two parts. The pairings in the two parts are independent since chords are not allowed to intersect. Thus, the $22$ points remaining are split into $8$ and $14$ points and you should read this MSE question about pairing points on a circle with non-intersecting chords.
Interestingly enough, that key idea is used to find the recurrence relation for pairing $2n$ points in a circle by non-intersecting chords. That recurrence relation is satisfied by Catalan numbers also and that proves the solution to the enumeration problem.
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