Way too Big to Count

Level pending

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.

The answer is 0.

1 solution

Kenny Lau
Jan 2, 2014

The marker needs to move at least 500 moves before the marker can reach a point at least 500 units away from the origin.
It can stop at any of the 500 dimensions, leaving you $1000-500=500$ moves to stop at.
The above two points give you $500\times500=250000$ possibilities already.
The 500 dimensions are all the same and interchangeable.
You'll at last have $250000$ multiplied by some whole number.

- The product will always have $\boxed{000}$ at the end.
Il faut que la marque mette au moins 500 pas avant qu'elle puisse atteindre un point qui est au moins 500 unités de l'origine.
Elle peut arrêter à tous les dimensions, et il a donc $1000-500=500$ pas possibles à arrêter à.
Les deux points au-dessus t'ont déjà donné(e) $500\times500=250000$ possibilités.
Les 500 dimensions sont toutes pareilles et interchangeables.
Tu auras enfin $250000$ fois quelque nombre entier.
Le produit aura toujours à la fin $\boxed{000}$ .

Way too Big to Count

The answer is 0.

1 solution

- The product will always have 000 \boxed{000} 0 0 0 ​ at the end.

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- The product will always have $\boxed{000}$ at the end.