A and B can complete a piece of work in 30 days, B and C in 20 days while C and A in 15 days. If all of them work together, the time taken in completing the work will be :
[A]$10\ days$
[B]$12\ days$
[C]$12\frac{2}{3}\ days$
[D]$13\frac{1}{3}\ days$

$\mathbf{13\frac{1}{3}\ days}$
Work done by (A + B) in 1 day $= \frac{1}{30}$
Work done by (B + C) in 1 day $= \frac{1}{20}$
Work done by (C + A) in 1 day $= \frac{1}{15}$
Work done by 2(A + B + C) in 1 day $= \frac{1}{30}+\frac{1}{20}+\frac{1}{15} = \frac{3}{20}$
∴ Work done by (A + B + C) in 1 day $= \frac{3}{40}$
∴ (A + B + C) will do the work in $\frac{40}{3} = 13\frac{1}{3}\ days$