A can complete a task in $8$ days while B can complete it in $12$ days. In how many days can C complete the task if all $3$ of them together can complete it in $3$ days?

1. Let the total work be represented by the $\text{LCM}$ of $8$ and $12$, which is $24$ units.
* Work rate of A $=\frac{24}{8}=3\text{ units/day}$
* Work rate of B $=\frac{24}{12}=2\text{ units/day}$

2. Given that A, B, and C together complete the task in $3$ days:
* Combined work rate of A + B + C $=\frac{24}{3}=8\text{ units/day}$

3. Calculate the work rate of C:

$$\text{Work rate of C}=(\text{A}+\text{B}+\text{C})-\text{A}-\text{B}=8-3-2=3\text{ units/day}$$

4. Find the number of days taken by C to finish the work alone:

$$\text{Days taken by C}=\frac{\text{Total Work}}{\text{Work rate of C}}=\frac{24}{3}=8\text{ days}$$