Find the minimum and maximum value of $\sum_{i,j,k} ({x_{ij} + x_{ik} + x_{jk}})$ given that $\sum_{i,j} {x_{ij}} = 225$ and $1\leq i<j<k \leq50$
My approach : i tried fixing k and getting the sum of ij term but that is not forming any pattern anyhow , next i tried making all possible combinations so that in that i can use the sum contraint given but that is not easy as we would need to plot in 3D , so is there a elegant approach for this ?
When you sum the three pairs across all triples, each pair appears $50-2=48$ times, so $$\sum_{i,j,k} (x_{ij} + x_{ik} + x_{jk}) = 48 \sum_{i,j} x_{ij} = 48 \cdot 225 = 10800.$$ That is, the minimum and maximum are both equal to $10800$.
In the linked question, this sum is not what is being minimized or maximized.
An equivalent way to think about it: in a complete graph on $50$ nodes, how many triangles contain edge $\{i,j\}$? One for each other node $k\not\in\{i,j\}$.
