$∣ ∣ ∣ ∣ \begin{matrix} a + b & a & b a & a + c & c b & c & b + c \end{matrix} ∣ ∣ ∣ ∣ =$

4 abc

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abc

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2 a^{2} b^{2} c^{2}

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4 a^{2} b^{2} c^{2}

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Solution

The correct option is A 4 abc
$∣ ∣ ∣ ∣ \begin{matrix} a + b & a & b a & a + c & c b & c & b + c \end{matrix} ∣ ∣ ∣ ∣$
$= (a + b) [(a + c) (b + c) - c^{2}] - a [a b + a c - b c] + b [a c - a b - b c]$
$= (a + b) [a b + a c + c b] - a^{2} b - a^{2} c + a b c + a b c - a b^{2} - b^{2} c$
$= a^{2} b + a^{2} c + a b c + a b^{2} + a b c + c b^{2} + a b c + c b^{2} - a^{2} b - a^{2} c + a b c + a b c - a b^{2} - b^{2} c$
$= 4 a b c$

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