$n$ balls are thrown randomly into $k$ bins - how many are empty?

We do the expectation, without finding the distribution. Let $X_i=1$ if Bin $i$ is empty, and let $X_i=0$ otherwise. Then the number of empty bins is $X_1+\cdots+X_k$, and the expected number is $E(X_1)+\cdots+E(X_k)$.

The probability Bin $i$ is empty is $\left(\frac{k-1}{k}\right)^n$. Thus $E(X_i)=\left(\frac{k-1}{k}\right)^n$. Multiply by $k$ for the expected number of empty bins.