$A$ and $B$ can do a piece of work in $10$ days, $B$ and $C$ in $12$ days, and $C$ and $A$ in $15$ days. The number of days taken by $A$, $B$, and $C$ working together to finish the work is

Let the total work be the LCM of $10$, $12$, and $15$, which is $60$ units.
* Daily efficiency of $(A+B)=\frac{60}{10}=6$ units/day
* Daily efficiency of $(B+C)=\frac{60}{12}=5$ units/day
* Daily efficiency of $(C+A)=\frac{60}{15}=4$ units/day

Summing all three equations:

$$2(A+B+C)=6+5+4=15\text{ units/day}$$

$$A+B+C=\frac{15}{2}=7.5\text{ units/day}$$

The time taken by $A$, $B$, and $C$ together to finish the total work is:

$$\text{Time}=\frac{\text{Total Work}}{\text{Combined Efficiency}}=\frac{60}{7.5}=8\text{ days}$$