Let A and C complete work in x days.

In 1 day,

(A+B) together can complete $\frac{1}{8}$^th of the work   ...(1)

(B+C) together can complete $\frac{1}{12}$^th of the work ...(2)

(C+A) together can complete $\frac{1}{x}$^th of the work   ...(3)

Then, (A+B)+(B+C)+(C+A) together can complete $\frac{1}{8}+\frac{1}{12}+\frac{1}{x}$^th of the work

∴ 2(A+B+C)$=\frac{5x+24}{24\times 2x}$

Given in the question (A+B+C) together can complete $\frac{1}{6}$^th of the work 

$\Rightarrow \frac{1}{6}=\frac{5x+24}{48x}\Rightarrow 30x+144=48x$

$\therefore x=\frac{144}{18}=8 days$

Hence option A is the correct.