Hello!

- Unless it is extremely obvious, you should try to justify where the formula came from.

- For P4: Take a look at the section about "Combinatorial Proof and Bijections" before the questions on the handout. There is an example about a natural bijection between the outcomes of flipping a coin $4$ times and the words of length $4$ made up only of the letters $H$, $T$: if you throw a coin $4$ times and get  heads, tails, tails, heads (in that order), then you can associate the four letter word $HTTH$ for that outcome. As Lucas Sha mentioned, you can see here there is a "clear pattern" there. In some way, it may even be silly  to say there is a bijection since it is clearly the easiest (natural) way to list the possible outcomes of throwing four coins. For the problem in question it may help to work on a particular example first, choose a small value for $n$ and list out all the numbers in set $A$, and all the elements in $P(B)$ (make sure to use the hint!). Be careful though, $P(B)$ is a set of sets. 

- For P5: you are correct, it is possible to have some empty boxes.

- On F4 there is indeed a typo, the problem should read 

$$\binom{7}{0} + 2 \cdot \binom{7}{1} + \cdots + 128 \cdot \binom{7}{7}$$

Best,

David