Way too Big to Count
A marker is placed at the origin in a \(500\)-dimensional coordinate system. The marker has \(1000\) moves, and each move consists of changing one coordinate by exactly 1. Find the last three digits of the number of ways the marker can get to a point that is at least \(500\) units away from the origin.
Details and assumptions
The marker has no restrictions on movement, it can move in any direction, forwards or backwards along any axis any time.
The marker can end with any nonnegative number of moves remaining.
The marker is forced to stop if it reaches a point at least \(500\) units away from the origin.