If a+b+c= 6 and 1/a + 1/b + 1/c = 3/2, then find a/b + a/c + b/a + b/c + c/a + c/b. Kindly show the detailed steps.

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a + b + c = 6

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{3}{2}

\frac{a}{b} + \frac{a}{c} + \frac{b}{a} + \frac{b}{c} + \frac{c}{a} + \frac{c}{b}

[\frac{1}{a^{3}} + \frac{1}{b^{3}} + \frac{1}{c^{3}}] = a^{3} b^{3} c^{3} .

a + b + c = 3, a^{2} + b^{2} + c^{2} = 6

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1

a b c

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} 1 & b & b^{2} 1 & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a)

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)

If a = 1, b = -2 and c = 3 then find the value of a $^{2}$ +b $^{2}$ +c $^{2}$ +ab+bc+ca.

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