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?

The problem with thinking about the number of choices for the first bin, then the second bin, and so on, is that each of these depends on the previous ones. You are right that there are $\sum_{i=0}^k\binom ki$ choices for the first box - this is the binomial expansion of $(1+1)^k$, so equals $2^k$, but then the number of choices for the second box depends on how many objects are left.
The right way to think of it is instead to consider the number of choices for the first object, then the second object and so on. For each object there are $n$ choices, and these are independent, so you can just multiply them together to get $n\times n\times...\times n=n^k$.
