For 17, 18, 19 and 20 try to identify first if you care about order and if repetitions are allowed. On 18 it is OK to use stars and bars since repetitions are allowed and we do not care about the order.

On 21, yes, you want the path to go through the letter C.

On 22 there should be $5$ possible ways of filling in the table as well. For each of the tables there is one path that can be associated with the table. Try comparing the $5$ paths with the $5$ tables.

For 23, remember that a bijection is a rule that matches the elements of two sets (in this case one is the set of all functions from $A$ to $B$ and the other is the set of sequences of length $4$ made up of $0$'s and $1$'s. So, we want to find a rule that tells us what sequence to write if we have a function from $A$ to $B$.

On 27 it might be good to start by looking at $6 = 1+1+1+1+1+1$. Any other possible sum could be obtained by removing some of the $+$ signs. For example you can think of $6 = 1+2+1+1+1$ as $6 = 1 + 1 \not + 1 + 1 + 1 + 1$.

On 30, none of them could stand at the ends (so the first and last persons in line are some of their friends).