Hey Derek!

For 26, try using complementary counting: how many regions would you have in a Venn diagram with three sets like the one described in the problem? Use this to find the total number of ways you could choose the sets $A$, $B$ and $C$. Then do the same assuming you only have two sets, this should help counting what happens when $A=\varnothing$.

On 27, try using PIE. Consider a set of numbers where $1$ is adjacent to $2$, and another where $1$ is adjacent to $0$. (Be careful not to accidentally count $6$-digit numbers that start with $0$, it might be a good idea to count when you have $01$ and $10$ separately).

For 4.29 try to proceed similar to example 4.9: Consider five sets that count the number of ways are there so that a specific person leaves with their own coat. Then use this sets and PIE to count how many ways there are so that at least one person leaves with their own coat.