`" if " |{:(1,,1,,1),(a,,b,,c),(bc,,ca,,ab):}|=|{:(1,,1,,1),(a,,b,,c),(a^(3),,b^(3),,c^(3)):}|` where a,b,c are distinct positive reals then the possible values of abc is`//`are

Options

A. `(1)/(18)`
B. `(1)/(63)`
C. `(1)/(27)`
D. `(1)/(9)`

Answer

A, B

Explanation

we have
`(a-b)(b-c)(c-a)=(a-b)(b-c)(c-a)(a+b+c)`
`rArr a+b+c =1`
As a , b, and c are positive
using `AM gt GM` we get
`(a+b+c)/(3) gt (abc)^(1/3)`
`:. Abc lt (1)/(27)`