On 24 we want to associate to each number in the set $A$ a subset of $B$. Note $A$ has exactly $2^n$ elements, and since $B$ has $n$ elements, there are also $2^n$ subsets of $B$ (for each element in $B$ it is either on the subset or it is not). The hint says "think of the elements of A written in binary", so for example to take $5\in A$, since $5 = 2^2 + 1 = 101_2$ and you can associate the subset of $B$ $\{1, 3\}$, since it has a $1$ in the first and third positions (from right to left) when written in binary. Can you do this with each of the numbers of $A$?

On 25 we want to show that counting the number of ways to put $n$ identical balls into $k$ different boxes is the same as counting the number of non-negative solutions to $a_1 + a_2 + \cdots a_k = n$. Can you think of a rule that assigns a way to arrange the balls with a solution to the equation (and vice versa)?