Consider the following board with a pawn in position $4$:
The game works by rolling a dice and going forward the number of positions as marked in the dice. So for example if you roll $1$ you go from position $4$ to position five. If you roll $2$ you get to the finish and the game ends. If you roll three however you end up at position $5$, because if you pass the finish you start going back.
However you can roll more than once. So for example, rolling a three and then a one will take you from position $4$ to position $5$ and then to the finish.
The question is how many different ways are there to get to the finish in $k$ or less steps (where one step consists in rolling one dice and moving the pawn the appropriate number of tiles).