In how many ways can $k$ distinct objects be placed in $n$ distinct boxes? In how many ways can $k$ distinct objects be placed in $n$ distinct boxes? Allegedly the correct answer is $n^k$ times, I just don't know how to arrive to this answer. In the first box I believe I have $\sum_{i=0}^{k}\binom{k}{i}$ options no? Then for the subsequent one I can't think of an expression. How should I go about this?

Consider functions from set $A=\{1,2,3, \ldots , k\}$ to set $B=\{1,2,3, \ldots, n\}$. Think of your objects as members of set $A$ and your bins as members of set $B$. Number of ways to do this assignment will be simply the number of function from $A$ to $B$. 
To count the latter, the first element $1$ has $n$ choices, the second element $2$ also has $n$ choices and so on. Thus the number of functions is $n \cdot n \cdot \dotsb n=n^k$.
