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A rook is in a corner of a two by two board and wants to get to the opposite corner in 100 moves. In how many ways can the rook perform this?

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Note: The rook may go through the opposite corner before the 100 moves, but needs to be at the opposite corner at move 100.

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There are

2^99 ways.

Each move, the rook switches colors. Each time, it can make two different moves. That is the case for the first 99 moves.

After move 99, the rook will be on a white square. To specifically get to top right, there will be only one move.

Addendum: OP added the clarification that the rook might pass through the top right before the 100 moves. My answer considered that as a given. But what if Rook could NOT go there? From the 99 moves mentioned, the 49 are white->black moves. These would not have two choices anymore, but one, so the final result would change to 2^50.

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