Grid Paths via an Intermediate Point

Path from $(1,1)$ to $(4,6)$: right steps $= 3$, up steps $= 5$. Number of paths $= \binom{3+5}{3} = \binom{8}{3} = 56$. Path from $(4,6)$ to $(8,10)$: right steps $= 4$, up steps $= 4$. Number of paths $= \binom{4+4}{4} = \binom{8}{4} = 70$. Total paths $= 56 \times 70 = 3920$.