Two obstacles via IE

Total. $\binom{8}{4}=70$.

Through $(2,1)$. $\binom{3}{2}\cdot\binom{5}{3}=3\cdot10=30$.

Through $(1,2)$. $\binom{3}{1}\cdot\binom{5}{2}=3\cdot10=30$.

Through both. Paths through $(1,2)$ then $(2,1)$? $(2,1)$ is reachable from $(1,2)$ by moving right+up. $\binom{3}{1}\cdot\binom{1}{1}\cdot\binom{4}{2}=3\cdot1\cdot6=18$.

Avoid. $70-30-30+18=28$.

Conclusion. $28$ paths avoid both obstacles.