All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?
Options are
- 49
- 65
- 70
- 96
Option 1 which is 49 is the correct answer.
Let the number of intersections be x.
We know that x <= 8C4 = 70, as every vertices on the octagon forms a quadrilateral with intersecting diagonals which is an intersection point. However, four diagonals intersect in the center, so we need to subtract 4C2 - 1 = 5. (70 - 5 = 65)
There are 8 type of three diagonals intersecting at the same point, so we need to subtract 8 x 2 = 16 (as we will count it once)
65 - 16 = 49
Another simple method using option
As we know maximum number of intersection is 70. there will be four diagonals intersect in the center, so we need to subtract 4C2 - 1 = 5. (70 - 5 = 65)Probably there will be more.
We have only 1 option which is less than 65.
i.e. 49
