Evaluating S depending upon following condition: Calculate the sum $S=Sigma Sigma Sigma x_{i} x_{j} x_{k},$

Suppose that $x_{1}, x_{2}, ldots, x_{n}(n>2)$ are real numbers such that $x_{i}=x_{n-i+1}$ for $1 leq i leq n .$ Consider the sum $S=Sigma Sigma Sigma x_{i} x_{j} x_{k},$ where summations are taken over all i, $j, k: 1 leq i, j, k leq n$ and $i, j, k$ are all distinct. Then S equals

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$S=sum Sigma x_{i} x_{j}left(L-x_{i}-x_{j}right) i neq j$

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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