If $a + b + c = 4$ and $a b + b c + c a = 6$ ., find the value of $a^{3} + b^{3} + c^{3} - 3 a b c$

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Solution

According to the problem:
Let the use $a^{3} + b^{3} + c^{3} + 3 a b c = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a)$
$= 4 (a^{2} + b^{2} + c^{2}) - 4 (a b + b c + c a)$
$= 4 (a + b + c) (a + b + c) - 6$
$= 4 \times 4 \times 4 - 4 \times 6$
$= 64 - 24$
$= 40$

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