Mathematics Asked on December 6, 2021
$S=sum Sigma x_{i} x_{j}left(L-x_{i}-x_{j}right) i neq j$
where $L=x_{1}+x_{2}+ldots+x_{n}$
$=L sum Sigma x_{i} x_{j}-sum Sigma x_{i}^{2} x_{j}-sum Sigma x_{i} x_{j}^{2}$
$=mathrm{L} sum mathrm{x}_{mathrm{i}}left(mathrm{L}-mathrm{x}_{mathrm{i}}right)-Sigma mathrm{x}_{mathrm{i}}^{2}left(mathrm{L}-mathrm{x}_{mathrm{i}}right)-Sigma mathrm{x}_{mathrm{i}}left(mathrm{M}-mathrm{x}_{mathrm{i}}^{2}right)$
where $mathrm{M}=mathrm{x}_{1}^{2}+mathrm{x}_{2}^{2}+ldots .+mathrm{x}_{mathrm{n}}^{2}$
$=mathrm{L}^{3}-3 mathrm{LM}+2 Sigma mathrm{x}_{mathrm{i}}^{3}$
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