Problem 8 · Count unordered groups by dividing out their orders · 5 points
Made by AI and not reviewed by a real person.
Nine dots lie on a circle, and every pair is joined by a straight segment. If no three segments meet at the same point inside the circle, how many distinct crossing points are there?
- 630
- 252
- 378
- 84
- 126
Show a hint
- Focus on one crossing and trace its two segments back to the dots that made it.
- Any four dots form a quadrilateral. How many pairs of its sides or diagonals cross inside the circle?
Show the answer and solution
Answer: E — 126
Each crossing comes from four dots. For any four dots, exactly one pair of segments crosses inside the circle: the quadrilateral’s two diagonals. Since no three segments meet at one interior point, different groups of four give different crossing points. Counting the dots in order and dividing out their 4 × 3 × 2 × 1 possible orders gives (9 × 8 × 7 × 6) ÷ 24 = 126. The tempting 378 counts all three ways to pair four dots, but only one pairing crosses.