Nonintersecting chords of a circle

We label 10 points on the circumference of a circle as \(P_1, P_2, \ldots, P_{10}.\) Find the number of ways to connect the points in pairs with non-intersecting chords, with no points left unconnected.

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Hint: When the number of points is 6, the number of ways is 5, as shown below.

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