White Dot, Black Dot

The diagram below shows a \(2 \times 10\) grid of dots (of negligible size) that alternate between black and white. How many ways are there to draw ten line segments between the dots such that

  • each white dot is connected to exactly one black dot (and vice versa),
  • no segment passes through more than two dots, and
  • none of the segments intersect each other?

Bonus: Generalize for drawing \(n\) segments on a \(2 \times n\) grid, where \(n\) is a positive integer.


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