<p>Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws <img alt="$25$" height="12" src="https://latex.artofproblemsolving.com/6/6/6/66644fd9f411299c3c1eabf151b5f601c9ef6c21.png" width="17" /> more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these <img alt="$27$" height="14" src="https://latex.artofproblemsolving.com/1/0/5/1051e7d41ec0b8bd688a872be17538413f4ee97b.png" width="20" /> line segments divide the disk.</p>

Question No: 410

Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these $27$ line segments divide the disk.

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Tags: Combinatorics Geometry Probability

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