Lattice paths forced through a power-up

A game character starts at $(0,0)$ and moves one unit at a time either right or up until it reaches $(6,6)$. How many such routes pass through the point $(2,3)$?

$350$. Count routes to $(2,3)$, $\binom52=10$, times routes from $(2,3)$ to $(6,6)$, which need $4$ rights and $3$ ups: $\binom74=35$. That gives $10\times35=350$.