What is the theoretical expected value of the sum of the highest 3 values of 4 fair dice? Many have estimated the expected value of the common "4d6, drop lowest" roll or averaged a complete enumeration of the possible outcomes. I would like to know the theoretical expression of this value, but can't wrap my head around the significance of dropping the lowest value.

Just to give a more computational approach than Andre, notice that what we want to calculate is
$$
E=\sum_{i=1}^6 \sum_{j=1}^6 \sum_{k=1}^6 \sum_{l=1}^6 \left(i+j+k+l - \min \{i,j,k,l\} \right).
$$
We can find this readily with various tools, such as PARI/GP, like this:
ss=0;
 for(i=1,6,for(j=1,6,for(k=1,6,for(l=1,6,
   ss=ss+i+j+k+l-min(min(min(i,j),k),l)))));
 print(ss/6^4)
