Monotone paths across a grid

MediumCombinatorics~4m

On a $6\times 6$ grid of cells you start at the bottom-left corner $(0,0)$ and move only up or right along grid lines to the top-right corner $(6,6)$ . How many distinct paths are there?

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Each path is $6$ ups and $6$ rights in some order.
Choose which of the $12$ moves are ups: $\binom{12}{6}$ .

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Monotone paths across a grid

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