Question

Sol. [Key Concept: $\sum_{0 \leq \ i < jle10}(i + j)$: count how often each index appears, or use $\sum_{i < j}(i + j)$ symmetry.]

$$i \rightarrow 10 - i,j \rightarrow 10 - j$$

$$\begin{matrix} & S = \sum_{0 \leq i < j \leq 10}^{}\mspace{2mu}\mspace{2mu}\sum_{}^{}\ (10 - i + 10 - j) = \sum_{0 \leq i < j \leq 10}^{}\mspace{2mu}\mspace{2mu}\sum_{}^{}\ (20 - (i + j)) \\ & S = \sum_{0 \leq i < j \leq 10}^{}\mspace{2mu}\mspace{2mu}\sum_{}^{}\ 20 - S = 20 \times 55 - S \\ \Rightarrow \ & \text{ }S = 550 \end{matrix}$$