$$P_{1}$$ and $$P_{2}$$ can do a piece of work together in 14 days, $$P_{2}$$ and $$P_{3}$$ can do the same work together in 21 days, while$$P_{3}$$ and $$P_{1}$$ can do it together in 42 days. How much work can all the 3 together do in 12 days?

Let's assume the total work is 42 units.

$$P_{1}$$ and $$P_{2}$$ can do a piece of work together in 14 days.

Efficiency of $$P_{1}$$ and $$P_{2}$$ together =$$\frac{42}{14}$$ = 3 units/day    Eq.(i)

$$P_{2}$$ and $$P_{3}$$ can do the same work together in 21 days.

Efficiency of $$P_{2}$$ and $$P_{3}$$ together =$$\frac{42}{21}$$ = 2 units/day    Eq.(ii)

$$P_{3}$$ and $$P_{1}$$ can do it together in 42 days.

Efficiency of $$P_{3}$$ and $$P_{1}$$ together =$$\frac{42}{42}$$ = 1 unit/day    Eq.(iii)

Efficiency of $$P_{1}$$, $$P_{2}$$ and $$P_{3}$$ together = $$\frac{Eq.(i)+Eq.(ii)+Eq.(iii)}{2}$$

= $$\frac{3+2+1}{2}$$

= $$\frac{6}{2}$$

= 3

Work done in 12 days by all three of them together = $$3\times12$$ = 36

Part of the work done in 12 days by all three of them together = $$\frac{36}{42}$$

= $$\frac{6}{7}$$