Answer to Question #333 in Combinatorics | Number Theory for Rashad

Question

Is it possible to divide 2n persons into pairs in 2n-1 different ways so that each person was paired with each other exactly in the same division?

Answer 1

Yes. Place one person in the center of the circle and arrange all others in a circle at equal intervals. Division into pairs will be conducted as follows. Choose one person standing in a circle, and draw a diameter through him. One pair is the selected person and the person in the center. All other pairs are those people standing in the ends of the chords perpendicular to the selected diameter.