P₁ and P₂ can do a piece of work together in 14 days, P₂ and P₃ can do the same work together in 21 days, while P₃ and P₁ can do it together in 42 days. How much work can all the 3 together do in 12 days?

$\frac{6}{7}$

P₁ + P₂ = 14 days,

P₂ + P₃ = 21 days,

P₃ + P₁ = 42 days,

⇒ Adding all the efficiencies, we get,

⇒ 2(P₁+P₂+P₃) = 6,

⇒ P₁ + P₂ + P₃ = 3,

⇒ Fraction of P₁ + P₂ + P₃ can do in 12 days = 3 x 12 = 36,

⇒ \(\frac{36}{42}\) = \(\frac{6}{7}\)