P, Q and R can do a project in 8, 6 and 12 days respectively. The project is started with P and Q working on day 1, Q and R on day 2, R and P on day 3 and so on. In how many days will the project be completed?

P, Q and R can do the project in 8, 6 and 12 days respectively.

So, P and Q can do in 1 day = $$\frac{1}{8}+\frac{1}{6}=\frac{3+4}{24}=\frac{7}{24}.$$

P and R can do in 1 day = $$\frac{1}{8}+\frac{1}{12}=\frac{3+2}{24}=\frac{5}{24}.$$

Q and R can do in 1 day = $$\frac{1}{6}+\frac{1}{12}=\frac{2+1}{12}=\frac{3}{12}=\frac{6}{24}.$$

After day 3 , work left = $$\left(1-\frac{7}{24}-\frac{5}{24}-\frac{6}{24}\right)=\frac{24-7-5-6}{24}=\frac{6}{24}=\frac{1}{4}.$$

So, P and Q can do ==> $$\frac{7}{24}\ $$part in 1 day.

or, $$\frac{1}{4}\ \ $$ part will do in = $$\frac{24}{7}\times\frac{1}{4}=\frac{6}{7}\ days.$$

So, Total time taken = $$\left(3+\frac{6}{7}\right)=\frac{27}{7}\ days.$$

D is correct choice.