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Consider a $3 \times 11$ rectangular grid as depicted in Figure $1,$ formed by $33$ tiles of area $1\text{m}^2.$ A staircase walk is a path in the grid which moves only right or up.

How many staircase walks are there from $\text{A}$ to $\text{B}$ which start by going to the right two times?

4 votes

We can only reach the grid point $(2, 0)$ by making two moves to the right in the beginning. Therefore the total number of staircase walks on this grid that start by moving right twice is exactly the same as the number of staircase walks from $(2, 0)$ to $\text{B}$ namely $^{12}\text{C}_{3}.$ In other words we need to choose $3$ up moves out of a total of $12$ remaining moves to make it to $\text{B}$ (the rest are right moves).

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