Grid Walking

Consider a grid. You start at coordinate \((0,0)\) and move one step at a time, eventually arriving at coordinate \((5,5)\). With each step you may move only one step east or one step north but never diagonally. How many paths are there from \((0,0)\) to \((5,5)\)?

Answer

You will take a total of ten steps. Five of these steps will be east; five will be north. You only need to choose which five of the ten steps are east. There are

\[ \binom{10}{5} = 252 \]

ways to make this choice.

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