If a + b + c = 6, a² + b² + c² = 30 and a³ + b³ + c³ = 165, then the value of 2abc is?

2

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(6)² = 30 + 2(ab + bc + ca)

36 - 30 = 2(ab + bc + ca)

(ab + bc + ca) = 3

Now, a³ + b³ + c³ - 3abc = (a + b + c) [a² + b² + c² - (ab + bc + ca)]

165 - 3abc = 6 [30 - 3]

165 - 6(27) = 3abc

3abc = 165 - 162 = 3

abc = 1

Hence, 2abc = 2