Monotone lattice paths across a six-by-six board

On a square lattice, a token starts at $(0,0)$ and must reach $(6,6)$. Every move takes it one unit right or one unit up. How many distinct routes are there?

$924$. Any route has $12$ moves, exactly $6$ of them up. Choosing which $6$ positions are the up-moves determines the route, giving $\binom{12}{6}=924$.