4.38 looks good. For 4.36, some of the +'s should be *'s right? Looks like you have the right idea.

For 4.30, some are missing, answer should be 11.

For 4.34, for the complement you're subtracting, don't we need to multiply by 3 to choose which group it is as well?

For 4.35, be careful with choosing the colors. For example, with 2 colors, couldn't we color 2 vertices each the two colors or alternatively color 3 vertices one color and the last vertex a different color?

4.39 is quite tricky. In my opinion I would worry about it less than the others, but it does have a nice trick. Expanding on the hint from last time, we can calculate the inner sum as follows:$$\sum_{B\subset S} n(A\cap B) = \frac{1}{2}\sum_{B\subset S} \left(n(A\cap B)+n(A\cap B^C)\right)=\frac{1}{2}\sum_{B\subset S} n(A)=2^{n-1} n(A)$$(Here we use the symmetry mentioned in the last hint.) The key is that the symmetry allows us to get rid of the dependence on B.

Hope this helps!