Question

# If ${a^2} + {b^2} + {c^2} = ab + bc + ca,$ then find the value of ${a^3} + {b^3}+{c^3}.$A. $3{(abc)^3}$B. 3abcC. $3{a^2}{b^2}{c^2}$D. None of these

$$\because {a^3} + {b^3} + {c^3} - 3abc$$
$= ({a^2} + {b^2} + {c^2} - ab - bc - ca)(a + b + c)$
$= {a^2} + {b^2} + {c^2}$
$= ab + bc + ca$ (given)
${a^3} + {b^3} + {c^3} - 3abc = 0 \times (a + b + c)$
$\Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 0$
$\therefore {a^3} + {b^3} + {c^3} = 3abc$