Lattice paths through at least one of two bonuses

On a grid you start at $(0,0)$ and must reach $(6,6)$ using unit moves to the right or upward only. Bonus items sit at the points $(2,3)$ and $(4,6)$. How many such paths pass through at least one bonus point?

$460$. Use inclusion–exclusion. Through $(2,3)$: $\binom52\binom74=350$. Through $(4,6)$: $\binom{10}{4}\cdot1=210$. Through both: $\binom52\binom52\cdot1=100$. Total $350+210-100=460$.