Suppose that we throw a die four times. Let $M$ be the smallest of the four rolls and let $S$ be the sum of the largest three rolls.
What is $E\;[M]$ and $E\;[S]$?
For $E\;[M]$ I suppose I could try and compute the distribution of $M$ and find the expected value directly, but there has to be an easier way. As for $E\;[S]$ I'm lost.
For $E(M)$, the procedure you mentioned is reasonable. Let us start the calculate. The probability that $M=6$ is $\left(\frac{1}{6}\right)^3$.
For the probability that $M=5$, note this occurs if all the tosses are $\ge 5$, but they are not all $6$. Thus $\Pr(M=5)=\left(\frac{2}{6}\right)^3-\left(\frac{1}{6}\right)^3$.
Similarly, $M=4$ occurs if the numbers are all $\ge 4$, but they are not all $\ge 5$. Thus $\Pr(M=4)=\left(\frac{3}{6}\right)^3-\left(\frac{2}{6}\right)^3$.
And so on.
When you find the expectation, something nice will happen.
Let $X$ be the sum of all four rolls. Note that $E(X)=4(3.5)$.
Note also that $X=M+S$, and therefore $14=E(M)+E(S)$.