Not Your Average Average

The average of all possible permutations of the expression $\alpha + \beta \gamma$ can be written as

$\dfrac{({\color{#D61F06} \alpha} + {\color{#3D99F6} \beta \gamma}) + ({\color{#D61F06} \alpha} + {\color{#3D99F6} \gamma \beta}) +({\color{#D61F06} \beta} + {\color{#3D99F6} \alpha \gamma}) +({\color{#D61F06} \beta} + {\color{#3D99F6} \gamma \alpha}) +({\color{#D61F06} \gamma} + {\color{#3D99F6} \alpha \beta}) +({\color{#D61F06} \gamma} + {\color{#3D99F6} \beta \alpha})}{6}$ $= \dfrac{ {\color{#D61F06} {\alpha + \beta + \gamma}} + {\color{#3D99F6} {\alpha \beta + \alpha \gamma + \beta \gamma}}}{3}$

This expression, by Vieta's Formulae, equals $\dfrac{{\color{#D61F06} 2}+{\color{#3D99F6} 4}}{3} = \boxed{2.}$

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Note that it doesn't actually matter what value $\alpha, \beta$ or $\gamma$ yields in the polynomial, since we don't need the exact value of $\alpha \beta \gamma = 8 - p(\alpha) = 8 - p(\beta) = 8 - p(\gamma)$ .

x = (2, 2 j, - 2 j}

2 + (2 j)(-2 j) = 6

2 j + (2)(-2 j) = - 2 j

-2 j + (2)(2 j) = 2 j

Therefore (6 - 2 j + 2 j)/ 3 = 2